NOt sure why you make a big distinction.
Legal and Illegal immigration flooded the labor market with tens of millions of extra workers.
And wages stagnated.
As to be expected according to the Law of Supply and Demand.
Yes, immigration put downward pressure on the wages of LOW SKILLED WORKERS. But overall, they have actually contributed the vast majority of what little wage increase we have gotten.
The Effect of Immigration on Productivity: Evidence from US States
The classic example is the brick layer. A low paid immigrant is hired to carry the bricks, now the skilled brick layer can spend more time actually laying the bricks while the immigrant carries them. That increases productivity and therefore the value of the brick layers labor. Now, if you want to look at immigration that possibly decreases the value of skilled workers I would direct you to the EB-3 Visa, which Trump actually wants to increase.