I'm in US History, so this would be about the past. I don't recall when exactly, but it's in history class, so I'd assume either 19th or 20th century, back when they worked in dirty and dangerous environments, low wages, but later than that, when they allowed it more, but then they started taking jobs and stuff.
closed as unclear what you're asking by Pieter Geerkens, Lennart Regebro, Kobunite, Tom Au, bhau Nov 25 '13 at 7:17
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The question contains a false premise. Immigrants didn't "take over" the country, the country assimiliated the immigrants, and the immigrants assimiliated into the country.
The false premise is that there is a distinct and independent class of immigrants whose interests are separate from the rest of Americans and incompatible. After all, the second generation aren't immigrants anymore, they're Americans. They can't take over the country; they're part of the ruling class, just like all other American citizens.
(Note Well: OP limited the question to the period of history where immigrants had a guarantee of civil rights. The answer is different during the period when the majority of immigrants were chattel. There are a variety of other related questions about the process of assimiliation that rapidly become both political and complex and are beyond the scope of H:SE)
Immigrants took over America multiple times. For example, immigrants that landed in Plymouth Rock and were welcomed by Americans soon decided that America belongs to them, slaughtered most Americans, and took their land.
And something vaguely like that, although not nearly as violent, happened quite a few times and keeps happening. Usually it goes like this: in about 2 generations the descenders of the immigrants forget their roots, assume that they are the true Americans, and start discriminating against the new wave of immigrants. Under the tremendous pressure the new wave of immigrants grows tougher and smarter and soon outperforms the "the true Americans" in everything that matters. That helps their grandchildren take over. And and in a couple of generations the cycle continues.
Example: Numerous "Irish and Polish need not apply" signs in New York businesses in the beginning of 20th century, but only 2 generations later an Irishman becomes the President of the United States.
This applies not only to immigrants, but to other minorities too. Another example: racist mobs in 1950s rule the South, but only 2 generations later a black person becomes the President of the United States.
You can see a newer wave of immigrants displace the descendants of an older wave of immigrants throughout much of American history. In fact, you can eyewitness the process yourself by comparing the maps of various "ethnic" districts with real life distribution of the corresponding stores and restaurants.