Applying to grad school where I've been on exchange
I know that many universities in the US and an increasing number in
  the EU are against taking on their own graduates


I assume you mean for grad-school ? This is the first I've heard of it. We encourage our undergrads to search widely, but I'd love to take my own undergrads for grad school if I could persuade them to stay. 

The sentiment you're referring to is more prevalent at the faculty hiring level: namely, universities are less likely to hire their own Ph.D students as faculty (at least not without some time spent elsewhere first)