Hear me out, Exchange citzenchip program. You can come live here if someone from this country wants to go live there. Then you exchange citzenchips. The population would remain the same and people could live in places they feel better at.
Now I'm curious on how the correlation actually lies, because it seems like it's actually correlated. When people can't enter, money is usually restricted too and vice versa, just like Russia is now. Of course there is smuggling, but it's for both money and people. Although maybe it shouldn't be money but "trade" in general.