I don't think that's the case with the USA, both the first colonies and immigrants from later centuries seemed to arrive purely based on the popularity of the american dream, being able to achieve fortune through effort or luck. It just so happened that this caused the huge cultural mixture that the USA is today.
I find the leftist hip teen open doors policy stupid as well, even when I'm an immigrant myself, because it's sold as a cute and colorful "There is room for everybody, regardless of race, gender and sexual preference" agenda, but it'd imply that if immigrants bring with the possibility of democratic vote they'd eventually want a mixture of islam and politics, which is called Sharia, and this particularly affect the gender and sexual preference, meaning this hip teen leftism is building it's own destruction.
This wouldn't exclusively apply to islam though, something I always say is that an example of what happens if you allow a mafia family to survive for centuries and centuries, there is a pretty big example, called Vatican.