One can move for economic opportunities or cultural opportunities. The two can co-occur but they are different needs. If we're talking about integration of Muslims, the ones who aren't soft on their own beliefs, i.e. not raised in the cosmopolitan West, are likely moving simply for economic opportunity or to avoid danger, not necessarily because they like the West.
Given that in most countries that are purely Muslim have reason to resent the West and see their struggle as being in part ideological, as Western materialist culture is antithetical to Islamic theocracy, should we not be a little bit wary?
I don't think culture is the only variable, but even if it was, there is still cause for concern.
No, which is exactly the sort of thing I'm talking about. "Free movement", to me, means that you are free to travel wherever you'd like on your own dime. It doesn't mean that you get to show up somewhere and demand citizenship, housing, medical care, or other free stuff. But obviously some people conflate these things.