The US takes in about a million legal immigrants per year [1], the EU took in 1.6 million immigrants from non EU countries in 2014 [2]. The US has a population of 320 million, and the EU has 508 million. So the immigration is about the same per capita. So I think it should be possible to deal with that.
[1] https://www.dhs.gov/immigration-statistics/yearbook/2015/tab...
[2[ http://ec.europa.eu/eurostat/statistics-explained/index.php/...