First immigrant generations and future generations
The term you’re looking for may be acculturation:
ac·cul·tur·a·tion (ə-kŭl′chə-rāshən) n.
- The modification of the culture of a group or individual as a result of contact with a different culture.
- The process by which the culture of a particular society is instilled in a human from infancy onward.
Another term, less specific to culture, is assimilation.
There are many studies of immigration, acculturation, and how immigrant groups maintain their own cultures and language.