Depends how you define 'mixing' or 'integrating' doesn't it ? The expat British in Antibes, when I had a place there, spoke little French and when they did it was execrable ! They mixed only with each other. In dealing with local and national government, they would seek the services of a French person who spoke English. There was, however, a supermarket, run by Britons, that stocked British goods and foods, lest the expats feared they were doomed to realise they were in France and felt homesick. French utilities did make some effort, in printing instructions etc in English. I suspect that, had the numbers been high enough, other government and quasi-government bodies would have done the same.
Does not all this apply to immigrants here? Except that many of them, out of necessity, do speak fluent English eventually, and those born of immigrants here all do.