Those wily immigrants are gonna destroy our culture!!!1
Sorry, but this is the most bogus argument fielded yet. If your culture/nation can't survive contact with an outside culture, that doesn't recommend your culture very much.

Strange, you never hear about omerta or xeer disappearing despite Italy being flooded with "rapefugees" or Somalia getting constantly invaded by Ethiopia. The reason why the immigrants are succeeding at all in steering the cultural narrative is that the current culture in Europe and the USA is markedly inferior to the fanaticism of Islamics and machismo of the Mexicans. At least they know the meaning of a hard day's work and already hate the government.

I'll admit, I'm not a fan of either Islamic or Mexican culture myself. Still, it's a darn sight better than the average Kop worshiping, social justice infected ninnie the public schools pump out. I'll also give you a hint about what leads to your 'culture' being inferior to even head-chopping suicide rapists. It is your obedience to authority. If you didn't have a massive warfare/welfare police state, foreign cultures wouldn't stand a chance of overtaking the indigenous culture. In fact, they'd be incentivised to integrate and adapt peacefully, as that's what free markets do.