According to LSE analysis, the Asian "concentrations" (as opposed to minorities more generally) voting pattern was actually rather consistent for Leave, across the UK:
Outside London, nearly every constituency with a double-digit South Asian population voted Leave. Luton has a 25 percent Asian population; Leave won there with a 19 percent majority. Places like Pendle, Oldham, Sandwell, Walsall and Wolverhampton also have high South Asian populations and voted Leave with large majorities. The only exception was Leicester, with its 30 percent Asian population – narrowly a Remain town, with a 2 percent majority.
While they don't have a formal poll in these areas, some anecdotal evidence does seem to point to Asians feeling pressure from EU migration indeed:
What about South Asians in London, a city that generally voted Remain? For me, the most interesting case is Newham, where I have lived for over a decade. Since 1991, the ethnic minority population has increased 128 percent and, at present, the South Asian population accounts for somewhere between 36 to 38 percent. Newham has a white British population of 16.7 percent – the lowest proportion in England and Wales. Remain won in Newham with only a six percent majority – slim for London.
For months before the referendum, everyone I spoke to in Newham – the local grocer; the Asian barbers; the chicken shop employees; the restaurants owners; estate agents; the underpaid workers; the tax-avoiding shop owners – supported Brexit. The arguments were the same: the rent prices, the NHS, the benefit cuts. The blame: immigration. More than this, there was the hope that once European migration stops, migration from South Asian countries can restart. It is a fight for resources between immigrants. Even though many first generation immigrants are overworked and underpaid, it’s better than the alternative. In the words of the British economist Joan Robinson: “The misery of being exploited by capitalists is nothing compared to the misery of not being exploited at all.”
Brexit means the loss of “freedom of movement”, but it is also a loss of European identity. [...] A loss of European identity doesn’t mean much to a South Asian barber. What about freedom of movement? You mean border controls? Well, “we are used to it”.
[...] Maybe it’s about fairness, then? A Bengali first generation immigrant, now a British citizen, told me that Brexit is “about equality”. What kind of equality? “Why is it that we have to suffer, pay thousands of pounds each year to extend our visa, face work restrictions, stay here for years, and only after that – after spending so much money and time – we get rights? And the Eastern Europeans get it straight away.”
Some pre-referendum interviews in the Guardian highlighted similar feelings:
Referring to himself as an Anglo-Pakistani, Mahmood is aware of the irony – coming from an immigrant family, he now wants to pull up the drawbridge on others. He says, however, that all his views are valid. “We have worked so hard to earn the right to live here and we contribute to the communities. What we don’t want is more people coming in who won’t bring anything positive and will just take,” he says. [...]
Research by the Runnymede Trust, a race relations thinktank, recently revealed that many BAME people are “ambivalent about the benefits of the EU”. A report said: “They appear less likely to take advantage of free movement [very few move about for work and arguably feel less ‘shared identity’ with others in Europe].
“Some view Europe in explicitly ethnic or racial terms, identifying fortress Europe as a way of keeping out non-white immigrants while allowing significant levels of.”
Interestingly and contrasting with these however is that overall
Two thirds (67%) of those describing themselves as Asian voted to remain
So there are probably some details that we're still missing (assuming the data/claim on "double-digit" Asian concentrations voting for Brexit is correct. I have some doubt on that because the data/claim isn't clearly about the Asian minorities in these areas, i.e. it could be that the white voters more overwhelmingly voted for Leave in these areas with "double-digit South Asian population"...)