One of main reasons people voted for Brexit was to restrict immigration. Is there actually a benefit to limiting immigration to the UK? Are there any studies available on this topic?
The BBC has a great overview of the impact of immigration on the UK. It uses a variety of sources and is in agreement with studies done on the subject.
Immigration has been a great benefit to the UK overall. It increases prosperity for all, creates new jobs, helps staff public services like the NHS without putting undue load on them, and contributes additional tax income. There is little evidence to suggest that it increases crime levels or has any other negative impact, beyond a little bit of pressure on jobs at the very bottom of the pay scale which is offset by the huge overall gains.
We must also not forget that in exchange for allowing EU freedom of movement, UK citizens benefit from being able to live and work in other EU countries.
Where there is possibly some detriment to the UK is the effect of unmanaged immigration on communities. For a variety of reasons some people dislike the nature of their communities changing due to immigration. Some of it is outright xenophobia or unreasonable demands such as not hearing any non-English spoken (it's unclear if Welsh and other regional languages are acceptable), but occasionally there are genuine complaints to do with acute problems or lack of integration.
However, since those issues could be resolved with some management, e.g. promoting understanding an integration, as well as doing more to tackle racism and bigotry, limiting immigration seems like the wrong solution to the wrong problem.
There are several broad groups of immigration, for which different approaches are required to ensure maximum benefit to the host country.
Refugees / asylum seekers
There are absolutely no benefits to taking in refugees voluntarily, as any benefit you receive from welcoming a refugee could likewise be obtained by welcoming a regular worker from elsewhere. Therefore the UK would benefit from completely shutting down all avenues for refugees to get to its land, which has been the official strategy for the past decades. The Calais Jungle existed precisely for this reason.
There is likewise no reason to tolerate illegal migration. If you need a certain kind of workers, you can issue them regular visas rather than letting random immigrants enter the country illegally. In fact, illegal migration results in a negative selection as law-abiding citizens stay in their home countries while those willing to break the law are entering illegally. According a Cato institute paper, legal immigrants are 60% less likely to be incarcerated than illegal immigrants.
Whether or not low-skilled migration is beneficial to the host country is disputed, but there is a general consensus that overall it brings more good than harm if managed properly. The UK has made the mistake of not requiring a 7-year transitional period when new countries joined the EU back in 2004 and thus experienced an unexpected surge of low-skilled migration, which upset certain segments of society.
In addition, the UK's laws for permanent settlement were too lax in past decades, which resulted in large communities of badly integrated second and third generation descendants of past immigrants. Thus the UK could benefit from tightening the requirements for becoming a British citizen, to ensure that only those who are able to fully integrate may permanently settle in the country.
There are pretty much no downsides to welcoming talented workers from across the world, presuming that their credentials are properly vetted. No one is complaining about foreign doctors or engineers settling in the country, as they raise the country's intellectual capital.
The UK attracts hundreds of thousands of international students from around the world. This brings money to the economy in the form of tuition fees, as well as additional intellectual capital. Students who graduate and find a highly-paid job can receive a highly-skilled employment visa and stay permanently. There are also numerous universities in the UK which provide sham degrees and the government is slowly cracking down on those.
Some countries get a significant number of people who wish to retire there. These are rarely controversial as retired people spend their money in the economy and don't resort to crime. The UK itself sees around a million of its own citizens living permanently in Spain. Of course, this presumes that the retirees either don't use up any public funds or that they come from a country that is integrated with your economy, like the example of Spain above.
If someone owns millions of dollars in assets and wants to come live in your country, you'd obviously welcome them with open arms. These could be both active investors and rich people who just want to live elsewhere.
Thus there isn't a clear cut answer to your question. Some groups should be restricted, other groups should be encouraged. Overall the UK is moving in the direction of improving the way it handles immigration flows, compared to how things worked in the past decades.
Framing the question here is very important. Certainly studies have been done, and I'm sure you'll see a lot of answers that include them. However, what you're asking and the statement made from the (presumably) other side are completely separate issues.
One of the main reasons people voted for Brexit was to restrict immigration.
This is firstly, a bit misleading. No developed country in their right mind cares about how many highly trained, highly skilled, working, professional or financially independent people want to come live, work, or invest in their country.
Almost all limits to immigration, and desires for them, come from imposing limits on people who are none of the above. The range of desire for reduced immigration can vary widely.
Even the most poorly/underdeveloped country in the world wouldn't encourage immigration from more developed countries if the only people who wanted to come were terminally ill/lifetime criminal/or disabled to the point of being unable to work.
Is there actually a benefit to limit immigration in the UK?
If we frame the question with the above in mind, the answer is no, but with caveats. Almost all developed countries have either negative or very small population growth rates. This means that they rely on immigration to even maintain neutral population growth rates.
However, that comes with the above caveat that some immigrants won't actually contribute in any meaningful way to your economy. For most of my lifetime, the bar has been set by bureaucrats based on predominantly economic factors. The idea of immigration being more of a right and countries having some level of civic duty to provide opportunities to potential immigrants is a fairly new(ish) idea.
Is there a benefit to limiting immigration to the UK?
If you consider higher wages (especially for the lower paid portions of society) a benefit then yes.
If you consider less unemployment a benefit then yes.
If you consider lower house prices a benefit then yes.
And besides those.
If you consider more training & advancement opportunities for the extant population arising from businesses inability to recruit abroad forcing them to train & promote from within a benefit then yes.
If you consider getting the countries population down to a level where it can support itself with food & power (or at least trying to) rather than relying on external sources that might conceivably be used to pressure it politically a benefit then yes.
What do you consider "a benefit" & who's perspective do you want the answer from?
In the interests of full Disclosure: I am one of the low paid so I do consider those to be "a benefit".
All those things can be considered benefits to one subset of the population but are at the same time also damaging to the interests of another subset of the population, so who's interests do you want to know about, the wealthy, the low paid, some other group?
Also the question can't be considered in isolation, restricting the pool of available recruits can lead to higher wages which may make UK goods less competitive so could lead to a loss of jobs, but if there are also increased tariffs the reverse may be expected (especially if you have a trade deficit).
Which now that I've got this far all leads me to suggest that.
Your question is much too broad & opinion based for any answer to be "the right answer".
Depends on the nature of the immigration. Most countries will welcome an influx of highly educated and skilled specialists wanting to join the work process. Fewer will welcome an influx of low-skilled labourers that seek to join the work process (if there is a labour shortage in the low skill sector, such immigration can prop up the economy, whereas if unemployment is already high, such an influx will only worsen the issue). But almost none want an immigration of freeloaders that have no intention of ever working and came to make a living by gaming the welfare system and commiting crime, regardless of what his skills or education are.
Of course, there is never just one kind of immigration at a time; all three are usually represented to some degree. In the case of UK, the third kind of immigration has been common enough to get a reaction out of the british people, especially the continuously worsening crime situation – rising knife crime has made headlines, for example.
Another issue, which is a bit of a modern taboo, is that with the combination of low birthrates of natives and high birthrates of immigrants, the native population faces the possibility of being entirely replaced over the following generations. They face the possibility of becoming a minority in their own country, their cultural values being replaced by newcomers who are showing no signs of seeking to integrate. Limiting immigration doesn't solve this issue (increasing native birth rates would), but it postpones the time when immigrants would reach majority status.
The combination of these concerns (crime, welfare, culture) thus creates the resistance to further immigration, hoping that doing so will prevent these issues from further worsening and perhaps provide some breathing room to attempt to integrate the immigrants.
As for whether there is benefit to limiting it, it depends on who's asking. A highly nationalistic person who cares chiefly for the cultural aspect will certainly say a loud YES. A person who cares nothing for that aspect and instead worries over who will work for his pension will probably say no, instead suggesting the first and second kind of immigration, the ones that seek to join the work process, be increased. The question is very general in this aspect, as there are as many definitions of what "beneficial" is as there are people in the world. Unless you are looking for a very general response, I suggest specifying that bit ("Is there an economical benefit", "Is there a security benefit", "Is there an ecological benefit", etc.)