The Turkic and Mongolian languages all came from the area around south Siberia and Mongolia. So ironically it is those Turkic people in Siberia who are the ones

The Turkic and Mongolian languages all came from the area around south Siberia and Mongolia. So ironically it is those Turkic people in Siberia who are the ones that stayed closest to home! The Huns might have been a Turkic group, but that's hard to prove or disprove since we don't have any records of their language beyond names and a few words that were probably borrowed anyway.