In 1-2 decades based on current stats
And in that phrase is the basis of that racist argument. The problem with the stat is though, that the birthrate doesn't stay the same for families of immigrants. There's endless evidence that first generation immigrants have birthrates similar to the country they came from, but within a couple of generations (and usually one) the birthrate becomes the same as the local rate. It's not even a new argument: people have been bleating about this non-stop for forty years or more. Variations of it go back hundreds of years. There's text from ancient Assyria where people are complaining about being out-bred by an influx of foreigners.
In answer to the original questions, yes, as people get richer they have less children. And yes, we should be grateful - the world needs fewer people, not more. The demographic time-bomb needs to addressed though.