Although I've lived in London twice, my viewpoint is obviously biased as an American living in New York City. But at least from over here, the "leave" campaign has not seemed to be based on a demand for reform of the EU [0].
I think that arguments for leaving the EU based on concerns about immigration are implicit calls for a rethinking of the EU social contract with respect to freedom of movement.
It's exactly this idea that discussions predicated on group identity are somehow outside the realm of legitimate politics that leads to the very resentment that made Britain leave the EU.
It's a shame, because if the EU had treated this issue as a political one, subject to compromise and regulation, they'd have been able to come up with a political solution. Instead, a lot of people treated this issue as a moral crusade and provoked a huge, destructive backlash.
Is there any real appetite for that in the rest of the EU, though? You need some kind of potential partners to put together significant reform of a 28-member bloc, and restricting the free movement of labour would be a huge change. I could see it happening if there are increasing questions in many EU countries about it, but not if there are only really increasing questions in one.
External immigration is a huge issue in many parts of the EU, and a common rallying point for the anti-immigration right. Especially Muslim immigration has become a major political issue in many EU countries. There is probably a coalition that could be put together with a hard-line view on refugees and external immigration (and one is sort of forming in opposition to the Merkel-led coalition's more accommodating view). But internal EU movement is a much smaller political issue in much of the rest of the EU (though it is a big issue in non-EU, but Schengen member Switzerland). Anti-immigration parties elsewhere in Europe, like Sweden Democrats, are mainly animated by worry about Muslim immigration from outside the EU, and don't really have the beef with e.g. Polish immigrants that some of the UK anti-immigration groups have.
Logically right, but emotionally wrong. I don't think you've appreciated the extent to which people publically oppose EU migration because it's politically incorrect to oppose Muslim migration, whether or not this UI indirection makes any sense.
[0] http://www.aljazeera.com/indepth/features/2016/06/brexit-ant...