Contrary to your previous claim immigration was never unlimited. It's impossible to compare the 1800s or even 1900s to today when travel was far more difficult. The Irish and others came here mostly on ships that took forever to make it to the states but most importantly they came legally. Today they're flying in by the millions, staying illegally or just crossing the borders as if they don't exist.
They came here "legally" because there was no such thing as "illegal immigration" back then.
Today they're flying in by the millions . . .
Yes, and the U.S. is a much larger country today than it was in 1850.