That's only true if you limit your definition of "cheap" to mean "cheaper than historical norms in geographies that have historically had high labor prices relative to those prevailing in migrants' original locations".

There's a common myth that immigration somehow reduces the cost of labor. This is nonsense. It reduces the price of labor in the location receiving the migrants (by increasing its supply) while simultaneously raising the price of labor in the migrants' original location by reducing such supply (both by direct labor outflows and by the increased leverage held by those who don't migrate, since they can now credibly threaten to do so).

Pretending that free migration is bad for workers is incredibly parochial and ignores the vast, vast majority of the world's workers, who currently live in places like India and southeastern Asia, and who would strongly benefit from the upward wage pressure that freer migration would bring about.