My first impulse was to harken back to the turn of the 20th century to the Irish and Italians and Eastern Europeans coming through Ellis Island. But it's now the turn of the 21st century, and immigrants from the ME, especially Islamics, have openly stated that they want to assimilate America, not be assimilated by her. The invasion analogy referring to "unassimilated immigration" was Pat Buchanan's I believe. Normally I'm loathe to quote him because I believe he's a self promoting crypto-fascist. But a government that invites 20 Mohammed Attas across its borders is no better than a government that would allow armed enemy invasion forces to camp out on the Mall in Washington "just to be friendly."
There's no practical difference between a country having an insufficient birthrate to replace its population and a country that can't field an army to defend itself. Both are doomed to be overrun and assimilated sooner or later.
A certain level of immigration from countries that have people who can be assimilated into our culture is good if the birthrate is too low. Overly rapid immigration isn't good and when the government interferes with assimilation it's also not good.