The current share of the population that is foreign born in both the US (13%) and Canada (a bit over 20%) is still not as high as that of the late 19th-early 20th century for the US (between 14-15%) and earlier 20th century for Canada (when it was also in the low 20s percentage-wise).
http://www.migrationpolicy.org/progr...tion-over-time
http://www.statcan.gc.ca/pub/11-630-...016006-eng.htm
Among other things, what's different are the source countries of immigration and the direction of flows. It's notable how many of the countries that were ones of net emigration (Germany, Sweden, UK, Italy, Greece etc.), now have become net receivers of immigration, some switching fairly recently. For instance, Italians and Greeks kind of stopped being notable immigrants to North America in the 1970s or so but a generation or two later, you see them now facing their own immigration issues, as with the contemporary crossings by migrants across the Mediterranean.
Also, people discuss the topic of brain drain from the developing world or the argument that people fleeing poorer countries should "stay and build their own country" rather than go to the west. But from the point of view of European countries the people were emigrating away from in the 19th century, the issue was similar and the loss in population was large too. During the Irish Famine, about a million people died and a million emigrated when Ireland's population was 8 million. Over a million Swedes emigrated away from Sweden during the later 19th and earlier 20th century, mostly to the US, which was near 20% of Sweden's population. These proportions are comparable to say, the contemporary Syrian refugees, who number five to six million according to estimates (most of who fled to nearby countries such as Turkey or Lebanon), of a population of 21-22 million in pre-war Syria, so about a quarter of people fleeing, displaced or emigrating away.
By contrast, the immigrant Mexican American population is about 12 million, the equivalent of ten percent of Mexico's population size, and Chinese and Indian emigration, often notable in Canada and in the US, while numbering in the millions is still small, even negligible compared to the total population of their sending country, each with over a billion, which goes to show how small of a relative proportion of them can actually make it to the west. African immigration to North America is still relatively small-scale and new, with about a couple million African-born Americans, and considerable less for Canada, since for much of the history of the New World, African migration was largely from the involuntary movement caused by the slave trade. So, all in all, despite some people portraying the 20th and 21st century Latin American, Asian and African immigration as massive and unprecedented, their numbers are not overwhelming in light of the tens of millions of people making up the European migrations to the New World over the 19th and 20th century.