I don't think countries should necessarily compete for the citizens of other countries, either, but they should have to compete for their own at the very least.

I don't think countries should necessarily compete for the citizens of other countries, either, but they should have to compete for their own at the very least. People aren't the property of their government, and if another city or state or country offers them a more attractive life than their home, they should be free to take that offer. The solution to emigration for a polity shouldn't be keeping their citizenry prisoner or forcing others to treat their residents more poorly because of some imagined 'race to the bottom', but to instead make their own land more attractive to stay and live in.