This is not a very common argument against immigration in general, but it has been made in the context of particular patterns of immigration. The argument is that immigrants from certain areas and institutions owe partial allegiance to the areas and institutions they came from. Conversely, those institutions exert control over the immigrants and interfere with laws in the target country of immigration with the aim (or pretext) of protecting the interest of these immigrants.
This argument has been made in the context of:
- Immigration from Mexico to the Southwest United States, up to and including the present. For more, see Mexican Reconquista.
- Immigration of Catholics to an overwhelmingly Protestant United States, at a time when the Roman Catholic Church projected a stronger image of international power and authority and a larger fraction of self-described Catholics felt a strong allegiance to the Roman Catholic Church.
There’s also Indonesia’s transmigration program–sending Javanese and others to demographically swamp the indigenous Melanesians of West Papua. It’s a deliberate ethnic/religious swamping program that has been described as cultural genocide. If Indonesia allowed these provinces to vote on independence, it’s doubtful that the vote would be successful, considering how many non-indigenous people now live in West Papua.