You have to toggle both knobs at the same time to make sure the system doesn't become unbalanced.
With trade+immigration, you can't allow the free movement of capital without also giving people the option of following their jobs to places long removed from their origins.
Is it practical? Not entirely, but the road has to flow both ways.
If you wanted to balance things, you'd pair an open market with shared rules on safety, environment, labor rights, etc.
The ole' america swircherroo.
As an aside, you don't even have to have large migration to produce a new equilibrium. As long as workers could feasibly migrate, they can negotiate for higher wages in their home country using that as a BATNA.
Americans are not migrating to third-world countries because there are more jobs. That's not going to happen.
But no, Americans are not emigrating in bulk any time soon. The thrust of my argument today is simply that greater inflows to the US is part of the 'free flow of labor' equation.