When I was in Mongolia, the story I got was that it was easy for them to immigrate to South Korea and mix in because they looked a lot like the South Koreans. Anyway, that was their story.
“Turkish and Korean are traditionally grouped under the Altaic language family, a classification that includes Turkic, Mongolic, and Tungusic languages, and often extends to Koreanic and Japonic families as “Macro-Altaic” or the Transeurasian superfamily.”
https://search.brave.com/search?q=language+group+turkish+korean&summary=1