An opponent gaining control of it is part of the replacement effect and happens immediately, even if you are in the middle of resolving the spell or ability and it has more instructions to perform after taking damage.

In some cases, a single spell or ability will deal damage to this creature multiple times during its resolution. In that case, a new player gains control of it each time, its ability triggers once each time, and the appropriate players control those triggers. After that spell or ability is done resolving, all of those triggers are put onto the stack in turn order, starting with the active player (or the next player in turn order if the active player isn't involved). Usually, this means that the cards are drawn in a different order than the control-changing effects took place.

You lose control of Khârn the Betrayer when it leaves the battlefield or another player gains control of it.

In contrast, the Sigil of Corruption ability is a triggered ability, and it won't go onto the stack until after the spell or ability that tried to deal damage finishes resolving.