The hunt for the suspect in the deadly attack on a Berlin Christmas market ended before dawn Friday in a shootout with Italian police near Milan, but the political fallout was just beginning to gather force.

Anis Amri, a Tunisian, lived in Germany for more than a year, despite having been previously jailed in Italy and denied asylum in both nations. Even as Europe’s most-wanted man, he traveled hundreds of miles this week, crossing at least two European Union borders on his way to Italy, where police stopped him to check his...