Immigration to the United States has always been as you say in the matter that many people who arrive here leave after a while. I used to have visions of nineteenth-century ships arriving full of immigrants to the United States, and then traveling empty back to Europe to pick up more immigrants, but the historical data actually show that about one-third of all persons who immigrated to the United States in the era of unrestricted immigration eventually migrated back to Europe. But they were changed after living here. Much of what makes Europe very livable these days results from returned would-be Americans promoting the democratization of Europe. (Much of the rest of what makes Europe very livable these days results from the United States rescuing Europe from central European tyranny twice during the world wars, and following the second rescue with the Marshall Plan and the Bretton-Woods agreement.)

So of course some people like to stay in the country where they grew up, where they are able to speak the language and are familiar with the cuisine and climate and where their childhood friends live. Nonetheless, the United States is second to no country in the world in its net gain of immigrants from various countries all over the world. Many, many, many people want to immigrate to the United States.[1] Russia is surrounded by wretchedly poor, badly governed countries that were formerly part of the Soviet Union, and many people leave those countries to go to Russia.[2] But Russia does not gain population through immigration to large degree, because many people who were born in Russia are glad to leave Russia (and they populate whole neighborhoods near where I live).

Prediction, as someone who has practiced immigration law in the United States: if the United States had open immigration again as it did until about the 1870s, it would gain a substantial percentage of population by immigration, even if some people who arrive return to their countries of birth. Usually people all over the world become more interested in living in America after hearing from other people who have lived in America, on a net basis.

[1] http://www.gallup.com/poll/161435/100-million-worldwide-drea...

[2] http://www.theatlantic.com/international/archive/2013/09/232...

AFTER EDIT: Ah, yes, the expected downvote again. Anyway, you are welcome to explain your disagreement, as I won't mind hearing a differing opinion from someone else here. That's something I learned in American culture: feel free to disagree, then hear someone out about why they disagree.