Dubai has managed to not let that happen even at rates far faster than replacement, but they don't give their immigrants voting rights, and usually not permanent residence (unless they are rich).
Right the only way to actually use immigration that way is essentially slaves and that has other economic (not to mention horrible moral) problems.
They definitely have literal slaves there, as well as the highest per capita rate of influx of high-net-worth individuals of anywhere in the world, and then a lot of people in-between.