A few instances of "new blood" in a population can be enough because it will then spread over generations.
In many countries the proportion of foreign-born is >10%, so the mixing is quite intense in terms of evolutionary time scales.
If that continues for a few hundreds of years, diversity will have significantly reduced. Indeed, some suggest that e.g. skin colour may largely homogenise within a thousand years.
A thousand years might sound like a very long time wrt. a single individual's lifespan, but for these issues it is nothing at all. In addition, some traits (e.g. pale skin colour) are quite recent from what we know.