"Back migration" from the New World would be interesting. Reindeer herding arose separately in both Scandinavia and Siberia, so what if the Athabaskans innovated it as well, say around the 1st century AD (the POD for my TL)? While I don't think you get my TL as a result, this innovation could cause the displacement of the Inuit from the Seward Peninsula and eventual migration into Siberia. If they get there early enough and are lucky enough then they could take most of the Chukotka for themselves (IIRC the Chukchi originated further south in Kamchatka). Obviously in the New World they'd migrate south as OTL so we'd have a similar (or likely greater, thanks to their reindeer herding) distribution of Athabaskans as OTL but with an additional group of Siberians.
This happened to a degree, they were just absorbed into the Malagasy and especially the Sakalava people.
Some theories hold there was a Yeniseian component in the Huns who ravaged Europe. It does seem plausible IMO for a Yeniseian group to advance into Europe (whether they survive a few centuries or are quickly absorbed by another group is a different matter).
The Evenks were too far away (they didn't spread northward into Siberia until later centuries) and Siberians is pretty vague. Which Siberian group, because from the Magyars to the Turks plenty of them did attack Europe.