If to be a "nation of immigrants" means that our ancestors came from somewhere else, every nation in the world (with the possible exception of East Africa, where the human race apparently originated), is a "nation of immigrants." Every nation in the world is populated by people whose ancestors came from somewhere else. The French didn't grow up out of the ground. The Japanese didn't grow up out of the ground. The Australian Aborigines didn't grow up out of the ground. Somewhere back far enough, their ancestors were immigrants. Even the American Indians are descendants of at least two waves of immigration from Asia. The fate of the American Indians ought to be a lesson on what happens when a nation doesn't control immigration.
Moreover, watch out for the term "migrants." The term is intended to imply that people have a right to "migrate" regardless of national borders, as if they were migrating Canada Geese. A nation that refuses to defend its borders is simply a geographical expression, not a people.
Just to make my situation clear, my father was an immigrant, my mother's parents were immigrants, and my wife is an immigrant. All came in legally. I welcome immigrants who come in legally and assimilate to our culture.