Some people reject arguments about rights and freedom. To these people, natural rights arguments ignore the fundamental importance of results. Humans don’t fundamentally care about means, they care about ends. That’s why they are called ends, Milton Friedman would say.
What are the results of immigration? In a capitalist society, people specialize in a certain profession and then trade their labor for the goods and services of other people. When an immigrant enters a profession, they compete with the people currently in that profession. If they provide a service at a lower cost, then people who purchase that service will have more money left over to purchase other goods and services. The total sum of human production will increase. These gains from immigration are equivalent to the aggregate gains from international trade.
Some utilitarians adopt a skewed view of utility. They value not increased productivity in general, but the increased income of the poor in particular. If immigrants enter the labor market, they may increase the total income of all Americans together, but competitive forces may decrease the incomes of the specific (poor) Americans that are most competing with immigrants. These concentrated losses are equivalent to the industry specific losses caused by international trade.
But immigrants moving into low wage industries in America are almost certainly moving out of even lower wage industries in other countries. Why else would they immigrate? A utilitarian ethic that supported immigration restrictions would value the well-being of poor Americans while ignoring the well-being of even poorer foreigners. This would be an obviously evil utilitarian ethic.
It may be inevitable for democratic processes to discount utilitarian gains to poor foreigners. But there is no reason for any individual utilitarian thinker to adopt the utilitarian constraints of their nation’s politics. Neither should a natural rights thinker accept the practical constraints of his political system as a moral constraint on natural rights. Justice is justice, whether or not it is procedurally obtainable.