What do you mean by their own power? You mean how the vex have time travel and such? The thing I that the Fallen's power derives from the traveler as well, and they just want to reclaim that power from us after their own abandoning. The Hive want to destroy the traveler to appease Oryx and the darkness, to win the war. This would mean the darkness wins. And they other two have other plans:the Cabal are more interested in studying the vex and I don't think they are really that knowledgeable about the traveler. And the vex also want to conquer in some artificial and emotionless way. They aren't going after the traveler but the traveler did stop their conquering so they may be pissed about that.