I have no doubt slavery would have ended eventually, but I seriously doubt it would have been a result of pressure from imigrants. Most likely by the 80's or 90's pressure would have come from foreign countries who consumed the South's cotton. The South learned a hard lesson when they discovered that Egypt easily picked up the slack for the world's loss of the South's cotton.
Slavery was only economicly viable when land is essentially free. A slave would only work as hard as he had to. A free man working his own land would be much more productive, and could bid a higher price for land