For the variacs, you can lubricate the mechanical parts but NEVER the brushes and the copper winding.
The brushes are made of graphite which is a lubricant itself.
+1 , only mechanical EXCEPT the exposed copper winding and the graphite brush contact that touching the copper.
For the exposed copper at the winding, if its too dirty or the oxidized copper is too thick, you can use a common rubber eraser for pencil to slowly & gently rub the exposed part, it takes time and it will clean them nicely. And do not touch as the grease from your finger will contaminate it.
Also DO NOT forget to blow off all those rubber pieces and make sure they're all gone completely once you finished.
I repaired & restored a Japanese variac while ago, bought used and lubricate all mechanical parts as they squeaked, and cleaned the copper using rubber eraser as above, now it works flawlessly and like new.
Few photoshots of mine when I tear it down, the black colored surface on the exposed copper actually was made by material transferred from the brush tip graphite material, and its fine & normal if it looks like that. Also its a mistake expecting to see it shine like buffed copper, as once the brush wipe that, it will turn black again and that black film will act like dried lubricant.