This gives most empires a degree of control over their population make-up without requiring them to be xenophobic or authoritarian. While in the early game you can simply choose not to have a migration treaty with a certain empire, later you find species migrating to you anyway by first migrating to an empire you have a migration treaty with.
Just do not allow that Species to migrate into your empire:
Migration Controls determines whether a species is allowed to freely migrate between worlds or not. Restrictions on migrations are always in place for slaves and pops that are being purged.
Setting thier rights (Citizenship, Livingstandart) low will also result in much less migration atraction:
In addition to setting rights for a species currently in your empire, you can also set rights for species outside your empire (for example granting species you would like to attract to your empire via migration Full Citizenship and a good living standard) and have a default set of rights that is applied to any species you have not specifically configured the rights for.
And Refugees can be controlled via "Citizen Species only":
Whether or not another empire is willing to accept those fleeing purges, slavery and resettlement depends on your Refugees policy. You can choose to accept other species will open arms, allowing refugee Pops to freely move into your empire, be more restrictive and accept only those Pops you have deigned to grant citizenship, or simply shut down acceptance of refugees altogether.
And you already can "purge" your coreworlds of specific species. What the sectors do should be of little relevancy for you:
The last thing we'll be covering today is some new policies that tie into the mechanics of species rights. The Core Worlds Population policy determines which Pops are allowed to live on your core (non-sector) planets, and can be set to either allow only citizen Pops (Full Citizenship/Caste System), citizen and slave Pops (Full Citizenship/Caste System/Slaves) or open them up to all species. If you restrict your core worlds and there are prohibited Pops living there, they will move away, either migrating to your sectors or fleeing your empire altogether if there is another empire willing to take them. It is also possible for Pops that are enslaved or targeted for extermination to escape your empire, particularly if there is an influential Xenophile faction that is helping them flee.
Have you tried to solve a problem that is already saved as part of the mechanics you wanted to expand?