Now that’s the kind of intelligent discussion I come here for!
According to the newspaper today, the museums in Sweden (and I’m guessing other countries) are preparing themselves for the dilemma of being offered artifacts from Baghdad, since on the one hand they don’t want to support the black market (and have an international agreement to do so) and the wish to return objects to the rightful owner (which they also have an agreement to try to do). It is likely that most objects will end up in the US and Switzerland to begin with, since they don’t have any regulations for that kind of thing, and the US likely has more money.