Immigration was one of the major issues in the referendum debate. The influx of several million Europeans coming to a country which had made no serious effort to accommodate its biggest ever increase in population changed the political landscape and enabled in no small way the decision to leave the EU.
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It's basically backwards. Hungary, Greece and Italy have a problem, not Germany. Merkel would be within her rights to ship people who show up in Germany back to those countries, but she's instead agreed for Germany to do a lot to help with it, and wants other EU countries to help too.