As others have said, it's already happening, and it'll only get worse. But since it's not western countries it's not highlighted much.

But when the AMOC stops and western Europe's winters get longer there will be huge changes too. If I recall correctly, the AMOC stopping is a trigger for an ice age, that is, ice sheets / the north pole going down way south. This would make anything above France uninhabitable, if not wiped off the map entirely.

But it'd be a steady process of increasingly cold winters, so likely in our lifetime it'd mainly mean we change how we build houses and buildings. But long term, people would move.