As for the Tuyuhun, they branched out of the Xianbei and the clan head was related to the head of the Murong Xianbei. Shiratori has reconstructed a number of these "Murong" words which appear to be both Mongolic and Turkic.
Paul Pelliot (1921) also reconstructed four Tuyuhun words based on their Chinese transcriptions. I'm not sure if he had Shiratori's (1913?) work at his disposal, but here are his proposals:
*agan 'elder brother' = cognates in both Mongolic and Turkic
*tšï 'your' = cognate to Mongolic *tši (2.s)
*kagan 'khan, emperor' = widespread term in NE Asia, found in Turkic, Mongolic, Tungusic, and even Koguryoic (cf. Beckwith
2004)
*makga 'father' = seemingly idiosyncratic
- Andrew Shimunek
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