IIRC, the other big factor is how wealthy the nation is. You probably have seen fears that immigrants will out breed the local populace in various countries. That would be true if they kept having the 5 kids or what ever is normal for where they are from. But if they moved to a wealthier country with a lower birth rate, their birth rates will align with the new country typically very quickly.
In a poor country, especially in a rural area, kids can be an asset and are, relatively, cheap to have. Move to a city in another coutry and they become a liability with their cost and upkeep jumping dramatically. And easier access to modern medicine doesn’t hurt either. And as you said, they have access to more education as well. Basically more options, and they will generally align their birth rates with the new country.
