people were immigrating from Europe

The linguistically correct term her would be emigrating from Europe.

to the US

This is immigrating.

emigrating to Europe

This is immigration.

from the US

The word you’re looking for is emigration.

emigrating from Europe

You’re correct here.

to the US

Once again, immigration.

immigrating to Europe

This is the linguistically correct use of the term.

from the US

Proper word would be emigrating.

Easiest solution is to say migrating

Migration by itself doesn’t indicate whether you’re referring to domestic-only movement, where people migrate from one city to another for instance, or domestic-to-foreign, or foreign-to-foreign movement.

It all depends on the boundary you set.

If your chosen boundary is Europe, people moving to Europe are immigrating there, and people moving from Europe are emigrating there.

If your chosen boundary is the US, immigration is moving to the US while emigration is moving from the US.

Since migration isn’t specific and can refer to any of the above cases, I prefer transmigration since “trans-” refers to “across” which I often interpret as “out from and in to”.

We don’t need to give up on prepositions in order to have more accurate language.