6.4 Spoilers
The chunk of Zodiark stuck on the Thirteenth's Red Moon would have been dissolved and its souls returned to the Lifestream when we (or Fandaniel / Amon / Hermes, I guess) defeated and destroyed him on the Source. Unfortunately, the Thirteenth's lack of a Lifestream due to being, well, the Void meant the souls had nowhere to go, and they ended up festering there until Golbez tosses Azdaja (as his Shadow Dragon) in there to create the monstrously powerful voidsent "god" Zeromus.
The fate of Hydaelyn's sundered self on the Thirteenth hasn't been explained, really. Her having sundered selves hasn't really been a crucial story beat, and I'm not convinced she actually did and just administered the sundered worlds from the aetherial sea. Regardless, her defeat and death should have sublimated her reflections' aether just like our own, assuming they behaved like Zodiark's.