people have this weird idea that foreigners come to Britain because of the benefits. This is about as likely as going to live in Australia (as many Brits do) because you like their dole system. It just doesn't happen. People move to Britain because they think, rightly or wrongly, that they'll get rich, and that means they plan to work. Most of them in fact do get work, because there are so many jobs around that Brits won't do (teaching, nursing, plumbing).
And of course the process works both ways. As people from poor countries move to Britain, Brits rush to buy cheap housing in their coutnries: Croatia is the latest, I believe. Perhaps, in the interests of nationalism, everyone who has a home in the Dordogne should return to Britain and take up teaching work.