The lazy American takes the immigrant's money and goes to live in the third world while the industrious immigrant gets to move to America.
There we go. Everybody will be happy.
We'll not overcrowd America, we get rid of our dead-wood, the immigrants get to come here, indolent Americans get a large sum of money to leave the country and go live in the third-world . . . problem solved!
Alternatively, we can offer prisoners the choice of going to prison or changing places with an immigrant from some third-world h*ll hole. Instead of paying the prisoner the immigrant pays his third-world government to make a citizen out of the American criminal.
Third-world governments would often be corrupt enough to go for the deal.