[eschaton]

Basically the only circumstance where I could see large-scale European immigration to the U.S. is if there was some ecological catastrophe. Like say global warming melts enough of the Greenland ice cap, causing the Gulf Stream to be pushed into deeper waters and not reach Europe. No Gulf Stream would, according to some climate models, chill Europe down to as much as the parts of North America on a similar latitude (Canada, more or less), which aside from being nasty to live through, would make it impossible for Europe to support itself via agriculture.

There's basically three things stopping European immigration from happening now.

1. Most of Europe is highly politically stable, meaning no refugees.

2. Europe as a whole has very low birth rates - particularly when you discount births by recent immigrants. Even if you're a country with a mediocre economy (like say Italy) when the number of 18 year olds entering the workforce each year is smaller than the year prior you're going to have enough job opportunities available domestically that there will be no reason to leave home. Indeed, refugee crisis aside, basically all Western European countries now have birth rates so low they basically need immigrants just to keep things going.

3. Most of the lower-income European countries are members of the EU, which means you can always, if you're a younger, underemployed person, migrate within Europe. This is why countries like Lativia and Romania are basically collapsing demographically - everyone young is leaving for Western Europe.

It's within the realm of possibility we might see another burst of immigration from Eastern European countries not within the EU (like Russia) similar to what we experienced in the early 1990s, but I wouldn't bet on it. Even in those cases, it's generally easier to try to migrate into the EU than get to the U.S. anyway.