The proper term is “migrate.” Birds undertake seasonal journeys between regions; this is migration.
“Immigrate” and “emigrate” apply solely to human movement into or out of a country, not to animals following seasonal patterns.
Migration, an instinctive behavior, aids birds in accessing food, breeding, and enduring challenging weather.
Using “migrate” accurately reflects the natural, cyclical movement characteristic of numerous animal species, especially avian ones.
Consequently, when describing seasonal animal movement, “migrate” is the most precise and suitable term.