A colony ship of unknown origins has come into existence in the space between universes with two unique factors to it: it possesses sentience, and it has the ability to replicate those from various worlds.
This is The Genesis.
An untold past event fractured the universe and the worlds in it, but it also damaged Genesis. Genesis has started churning out replicants from other worlds to help repair itself and to help it with a mission: to help return it to its world, while fixing the universe along the way. Replicants are created using the DNA and memories of their canon world counterparts. So while they exist on Genesis, their original selves are still going strong in their home worlds.
Why would they bother staying on this ship when they could freely leave the second it touches down on another world anyway? Well, the universe will continue to fracture and fall apart until it implodes if The Genesis can't get back to its world to defeat FRED - a nebulous being intent on destroying the very fabric of existence itself.
So fix Genesis, save worlds, and restore balance because if you leave, Genesis is doomed to fail its mission.