I don't know if the authors of the paper "Why Is There No Race to the Bottom in Capital Taxation?
" actually support supply side economics, but their argument as to why we don't see a race to the bottom in practice comes down to:
Governments thus face a trilemmatic choice when choosing a tax system. They
can simultaneously reach only two of the following three policy goals: (i) maintain a solid capital base; (ii) generate sufficiently high-tax revenue; and (iii) avoid social inequity by treating different incomes differently.
In (iii) they refer to taxing mobile capital vs. (e.g.) immobile labor substantially differently.
They argue that (ii) and (iii) are limited by the electorate/democracy, e.g.
voters
reduce political support when governments run large deficits, cut spending
on relevant issues, and implement policies which voters perceive as unfair.
They claim to have empirical data to support their model (across OECD countries)... I haven't looked into that.
Also, they don't seem to talk about what could happen in countries with a large democratic deficit. Presumably one (or both) of the second or third pillars could be sacrificed... at least temporarily. I guess some non-electoral means of correction would eventually kick in: revolution etc. or at least massive tax evasion by the defavorized groups (e.g. by labor; actually that kinda used to happen in Eastern Europe, though perhaps for different reasons). Oh, and zero CIT (corporate income tax) apparently happened in Moldova in 2008. Apparently Moldova reintroduced it (at 12%) in 2012. There might be an interesting story there...
And since Tim reminded me of the Apple in Ireland case.... it's probably worth mentioning the fairly awkward steps on tax coordination/harmonization at EU level (which is another [potential] counter-mechanism):
[Some terminology first:] Tax coordination refers to a cooperative
tax setting, where countries or a group of them build on domestic tax systems to render
them compatible with the aims of the Union as formulated in the TEU. Consequently, countries
deliberately give up parts of their autonomy in tax matters. Harmonization is viewed as
tighter coordination, leading to almost identical or at least similar tax systems, tax bases and
tax rates within a Union. [...]
A race to the bottom due to increased tax competition was feared since the very beginning of
European integration. It was one of the driving forces behind any attempts to harmonize taxes
at the European level. However, tax harmonization succeeded only moderately so far. The
main reason was that Member States did not agree on the necessity and also the scope of
harmonization, apart from the fact that tax matters remain one of the few areas where a proposal
needs unanimity to pass the Council of the EU.
In particular, Ireland and Hungary currently oppose corporate tax harmonisation. (There's a higher level of VAT harmonisation in EU though, with a minimun rate agreed.)
Again, I don't know what proponents of supply side economics (in general) think of tax harmonisation... but if we admit Macron is one of them, then I should mentiond that he is [at least] in favor of a Fraco-German corporate tax harmoniation, perhaps becuase a EU-wide corporation tax harmonisation doesn't look polliticaly feasible at the moment.
And the smaller fish can sometimes be goaded into compying at least paritially, as in the case of EU-Switzerland relations:
Switzerland has come under intense pressure from the European Union (EU) and the Organisation for Economic Cooperation and Development (OECD) to change “harmful” tax practices in the existing tax regime. Most notably, cantons currently charge some foreign multinationals lower rates for profits derived outside of Switzerland.
Following a consultation period, the Swiss government released the final version of its latest reform proposal on Wednesday. It attempts to steer a path between meeting international tax standards and retaining the 4,000 foreign firms that enjoy breaks under the existing regime.
‘Tax proposal 17’ calls for at least 30% of company profits to be fully taxed by cantons before so-called patent box and other research and development reliefs can be applied. Foreign companies that currently enjoy tax breaks would end with a larger bill while smaller firms would pay less, according to a government statement. [...]
The Swiss Business Federation (economiesuisse) said it was crucial that the “most important tax reform in decades” is passed at the second attempt. The cost to Switzerland of a second rejection would be “simply too high”, the lobby group stated. [...] In the worst-case scenario of failing to reform corporate taxes, Switzerland could face sanctions from the EU and OECD.
Last year, voters threw out the initial tax reform plans that were deemed too complex and favoured corporates at the expense of ordinary tax payers. To help convince voters in a potential new referendum, the government has pledged to increase child allowances by CHF30 per child if the package is adopted.
The last paragraph sounds like point (iii) [from the fist paper I mentioned] in action (although it does sound like a fairly lame "bribe").
And since OECD data [again in the first paper discussed] may be too restrictive, there's an IMF paper on wider set of countries:
This paper assembles a new dataset on corporate income tax regimes in 50 emerging and
developing economies over 1996-2007 and analyzes their impact on corporate tax revenues and
domestic and foreign investment. It computes effective tax rates to take account of complicated
special regimes, such as partial tax holidays, temporarily reduced rates and increased investment
allowances. There is evidence of a partial race to the bottom: countries have been under pressure
to lower tax rates in order to lure and boost investment. In the case of standard tax systems (i.e. tax
rules applying under normal circumstances), the effective tax rate reductions have not been larger
than those witnessed in advanced economies, and revenues have held up well over the sample
period. However, a race to the bottom is evident among special regimes, most notably in the case
of Africa, creating effectively a parallel tax system where rates have fallen to almost zero.
Regression analysis reveals higher tax rates adversely affect domestic investment and FDI, but do
raise revenues in the short-run.
I guess this (race to the bottom under special regimes but not under normal ones) is one the reasons why the EU apparently hates special regimes.
And regarding the US, at least at inter-state level, there's one paper by two Chicago economists (your guess if they are supply side advocates) concluding that:
Our results suggest that the secular decline in capital tax rates, at least among U.S. states, reflects synchronous responses among states to common shocks rather than competitive responses to foreign state tax policy. While striking given prior empirical findings, these results are fully consistent with the implications of the theoretical model developed in this paper and presented elsewhere in the literature. Rather than “racing to the bottom,” our findings suggest that states are “riding on a seesaw.” Consequently, tax competition may lead to an increase in the provision of local public goods, and policies aimed at restricting tax competition to stem the tide of declining capital taxation are likely to be ineffective.
And OECD has a paper that has among its conclusion has a list of (non-central) taxes sorted by the order in which they experience downward tax pressure:
The intensity of tax
competition depends on the tax category households and firms are subject to. Sub-central
business taxes, i.e. taxes on capital and capital income, are most prone to tax base mobility and
hence tax competition, followed by sub-central personal income taxes. Taxes on residential
property are the least affected by tax competition. Competition on consumption taxes depends
strongly on the size of jurisdictions.
The same paper also lists some mitigating factors, mainly size and diversification of the local economy:
Several other factors affect tax competition and tax base mobility: Large jurisdictions with
agglomeration economies are less affected by tax base mobility and tend to set higher tax rates.
Capitalisation, i.e. the impact of taxation on property prices, reduces tax base mobility and tax
competition. Spending on public services, especially infrastructure and education may
compensate for higher tax rates and reduce competitive pressure. Fiscal equalisation reduces tax
base mobility and tax competition.
Although this paper is about sub-central taxes, I think it's safe to infer from it some global conclusions: for instance the corporate income tax experiences the highest pressure (among the nation-level taxes), but we don't see widespread zero corporate taxes among large countries. Instead only some small "fiscal paradises" have zero corporate tax.
And a 3rd worthwhile point from the OECD paper (I'm giving you the expanded version found later in the paper)
SCG [sub-central government] tax rates tend to go up rather than down, and they tend to converge over time, regardless of
the tax type (Figure 5). A “race to the bottom”, especially between wealthier SCGs, cannot be observed.
This tends to contradict the view that tax competition may result in taxation levels too low to sustain
adequate public service levels. Moreover, the trend for rising tax rate differences between SCGs cannot be
confirmed: for the countries under scrutiny, tax rates mostly tend to converge rather than to diverge. While
the results do not cover all OECD countries with highly autonomous SCGs, they nevertheless provide a
fairly broad sample of SCG tax setting behaviour.
Different factors may explain why the “race-to-the bottom” hypothesis is not confirmed. First, in
many countries where SCGs tap the same base as central government, vertical tax competition – which has
the effect of increasing rather than decreasing tax rates – may countervail horizontal tax competition
forces, as is best shown by the example of France [...] but is also evident in more decentralised
countries such as the United States or Switzerland. Second, in some countries SCGs are not allowed to
lower tax rates below a certain threshold, so are actually prevented from further competing on tax policy
once the bottom has been reached [...]. Third, many fiscal arrangements, particularly fiscal
equalisation can actually reverse incentives and make SCGs increase rather than decrease their own tax
rates, with a trend towards more equal taxation across SCGs [...].

I'm rather underwhelmed by this last graph because all data is from before the financial crisis (this might be on purpose though) and using the sales tax in the US is perhaps less representative than what Tim mentioned as the main problem there (local tax breaks on investments)... the fact that that tax would experience higher pressure is actually acknowledged by the previous conclusions of the OECD paper. I suspect there might not be enough quality data for a comparison though, given the difficulty with accounting for all the special local tax deals.
Let me not forget their (OECD) points about vertical tax competition (which has an opposite effect, of increasing taxes):
Vertical tax competition occurs when different government levels have individual discretion in setting rates on a
common tax base. When an individual government or government level changes its tax rate, it affects the tax base for
other government levels. For example, an increase in a central government capital tax reduces national savings,
thereby reducing the capital stock in all sub-central jurisdictions. Similarly, an increase in the central government
personal income tax reduces incentives to work and hence the income tax base for all SCGs. Since the tax increase
imposed by one government level diminishes tax revenues for the other government levels, these in turn may have to
increase their own taxes in order to rebalance budgets. The tax base becomes a common good, where each
government level is imposing a tax externality on the others. Vertical tax competition can be quite pervasive in
countries with concurrent taxation of corporate income, personal income or sales and turnover taxes. Examples are a
central government income tax on which SCGs set individual surcharges or a combined central/sub-central VAT/sales
tax.
Vertical and horizontal tax competition interact. Vertical tax competition tends to raise tax rates and hence to
partially offset the effects of horizontal tax competition, but the overall effect depends on the tax mix and the elasticity
of the shared tax base. Over-taxation might become an issue if an inelastic tax base – such as the property tax or
some consumption taxes like the gasoline tax – is shared across government levels, while it could be less salient when
a more mobile base – such as when the corporate or personal income tax – is shared. Vertical tax competition also
depends on the extent to which the central government can commit as a “first mover” to a tax policy that SCGs then
take as given. In other words, the more “hierarchical” the relationship between the central and the sub-central level, the
less significant is vertical tax competition. Finally, government’s objective functions in terms of taxation as well as
political economy constraints – such as voter’s behaviour – limit the extent to which government levels can exploit the
joint tax base. Tax policy coordination across government levels may further help reduce vertical tax competition and
excessive taxation.
As a coda, since most of the above is fairly "mainstream", I tried (at K Dog's suggestion) to find some Austrian (economics) school views on this issue. Mises Institute has an article titled "race to the bottom", but that article talks only of election promises. Separately Mises Institute has another article finding nothing wrong with tax competition:
For a supporter of Misesian theory, any mechanism that puts pressure on governments to lower taxes has to be good news. Often the only incentives for government activities and spending seem to be upward ones, as lobby groups and politicians seek to extend the power of the State. Hence the historical trend for taxes to increase, and anything that helps counter this will be beneficial.
Tax competition does not prevent citizens, through electing their governments, from setting whatever tax systems they prefer. However it does mean that countries will face the consequences of their choices, and it gives an exit mechanism for minorities to protect themselves from victimization. It also helps to counter the democratic deficit caused by the political class offering a very limited choice of policies that does not reflect the range of the electorate. Unfortunately this makes tax competition as popular amongst governments as other forms of competition are amongst large corporations, and so often their reaction is to form a cartel to guard their patch from upstart smaller competitors.
So basically, they deny that tax competition creates any problems that the market can't fix by itself, so no worries. They go on to lambast any tax harmonisation as a "cartel", using EU as an example.
Apparently I managed to find a self-described supply sider, namely Dan Mitchell, who also [co]wrote a 2008 book titled Global Tax Revolution: The Rise of Tax Competition and the Battle to Defend It. And in this book (p. 8) we read:
The [...] claim is that tax competition creates distortions in the
public sector. Any reduction in government revenue that results
from capital and labor’s emigrating to lower-tax nations is supposed
to be an inefficient ‘‘fiscal externality.’’ Government revenues will
fall below the supposed optimal amount as a ‘‘race to the bottom’’
in tax levels ensues.
In Chapter 7, we discuss the theoretical flaws in these arguments.
For one thing, they are premised on the ‘‘public interest theory of
government,’’ the idea that government officials always act for the
general welfare of citizens. We argue that it is naive to assume that
if policymakers had monopoly fiscal power without tax competition,
they would set tax rates at the optimal level for the good of the
people.
At a practical level, there has not been a race to the bottom in tax
revenues around the world, as the critics fear. We wish that there
had been, but tax competition has not yet ‘‘starved the beast’’ of
bloated government. However, in most advanced economies, tax
revenues as a share of gross domestic product have leveled out in
recent years, and even declined in some places. We also think that
governments would have grown larger without the rise in tax
competition.
In chapter 7 they mostly talk about how OECD's attempt to define unfair tax competition is hopelessly flawed by complexity. The also argue that--by analogy with free trade--tax competition is good for global growth. And they have one point that includes migration:
A final note regarding labor flows is needed. It might appear
that the tax-induced migration of skilled workers, or brain drain, is
indeed a zero-sum global game. But when skilled people move to
countries with lower taxes, they will likely increase their work effort
and put their brainpower to better use. It is true that brain drain
has a substantial effect on poor nations, but even here experts note
that emigration has benefits. Emigrants send large financial remittances
to their home countries, which is an efficient form of aid. Also, emigrants often return home with new skills that they can use
to help their countries. Finally, the emigration of skilled workers
provides a strong signal to governments that they need to reform
their domestic policies.
So the arguments in this book seem mainly that tax competition is mostly good (theoretically) and that the (theoretically) bad aspects of it have yet to materialize... This position seems to be somewhere in-between the Mises-style theoretical view and the mainly empirical results discussed in the largest part of my post.
Unlike Mises-style arguments, this book doesn't argue that low taxes are always desirable. You may wonder how is it that supply-side economics; well it depends how you define that term. This is how Mitchell defined it (not in the book, but in that email exchange published in The Atlantic:
I also would augment your characterization of strong, semi-strong, and weak versions of supply-side economics by adding the "somewhat-strong" version, which is that changes in marginal tax rates affect incentives and lead to changes in taxable income that can have a measurable impact on economic performance. I think it is fair to say that this is what most "supply siders" actually believe (which, to be sure, is different than what some overly-optimistic Republican politicians assert).
And this is what they say in the book about tax policy diversification:
Countries with different industrial structures may favor different
tax policies. A country that has a large tourism sector, for example,
may want to have low taxes on incomes, but higher taxes on hotel
rooms so that foreigners pay some of the costs of government services.
Another example of tax diversity was suggested by economists
Richard Baldwin and Paul Krugman. They argue that tax harmonization
would be inefficient in Europe because ‘‘core’’ and ‘‘periphery’’
countries on the continent should have different tax structures.
The University of Chicago’s Julie Roin suggests another situation
where tax diversity makes sense. Residents of wealthy cities may
not want any new factories because of the resulting pollution, and
they will not favor tax cuts to attract investment. However, people
who live in cities with high unemployment would be happy to
provide tax cuts to lure new investment.
However, it might be that tax competition leads to tax systems
that are more similar over time, an evolution that can be called
‘‘competitive harmonization.’’ That appears to be what is happening
across major industrial countries regarding corporate tax rates. Rates
have been cut in recent decades, but they are also in a narrower
range than they used to be. Thus, the concerns of harmonization
advocates have been partly met, but the harmonization is occurring
at a lower rate level than advocates would probably prefer.
So yeah, self described supply-side economists citing a Krugman paper. Heresy!