Consider an insectoid race where each worker/soldier/role is bred as required by the queen. This is very similar to ants on earth. Periodically new Queens are born who then set out to found their own colonies.
We've discussed what happens when a starting population's gene pool is too small. However on earth colonies using this model clearly exist and thrive.
In a civilisation where every "minion" in a colony is bred from the queen (and presumably her own offspring) how does a colony survive past it's first few generations and avoid suffering from genetic issues caused by having a small gene pool or a single individual?