It does vary, of course, and some of the top schools also have a variety of programs to choose from. But in my experience elite schools in the US also partner with elites in Europe. For example, the only way to study abroad as an American undergraduate at the Ecole Normale Superieure (the top school in France in literature, philosophy and math) is to attend Harvard, Yale, Princeton or one of a handful of other top schools. (And even those top schools are only allowed to send 1-2 people per year, per ENS’ restrictions.)