I was consulted with this question and I found myself somehow confused:

let's say I gain control of my opp's Darigaaz Reincarnated (with a Hijack, for instance), then it dies and I exile it with three egg counters.
Does Darigaaz:
A. Triggers during owner's upkeep and reincarnates under their control because it becomes a new object on exile, or
B. Triggers during my upkeep and reincarnates under my control because I control the effect the exiles it?

Darigaaz Reincarnated will trigger during its owner's upkeep; the triggers will be controlled by Darigaaz's owner, and its owner will be the player who puts it onto the battlefield once all the egg counters are gone.

Cards in the exile zone are not controlled by any player, since in a normal game only emblems plus things on the stack and battlefield have controllers. (CR109.4) And since Darigaaz Reincarnated doesn't currently have a controller, the words “you” and “your” on it refer to its owner, so that's the player whose upkeep it's looking for. (CR109.5) When its ability triggers, that ability will be controlled by Darigaaz's owner, since Darigaaz has no controller. (CR112.8) And of course, the controller of a triggered ability is the player who carries out its instructions. (Unless specified otherwise, which isn't happening here.)

It doesn't matter who controlled Darigaaz when it was exiled, nor who controlled the effect that caused it to be put into exile in the first place–all that matters is that it is in exile, and has an egg counter on it.