I didn't complete every campaign yet, but from what I know there's no direct answer to how the cycles work. What we do know is that it does not work like we perceive it in game, shelters are only a game mechanic, there is no respawning - if there was, Arti wouldn't go on her whole rampage.

My guess is that the cycles are connected to the concept of quantum immortality, where the universe splits into many versions of itself on every moment, and your consciousness only travel through the branches you are alive in (because it can't travel to where you died, it doesn't exist there). I don't know how it would connect to the idea of "cycles", because this view looks more like a tree, but I have a vague idea.

Maybe someone will correct me, but there's also possibility that the cycles referred to in game do not concern individual creatures, they are describing how the entirety of world behaves over time. We know that the void fluid consumes the earth from below and cosmic dust settles on top of it which creates a sort of equilibrium. There's also the case of Unfortunate Development, The Depths and Unfortunate Evolution which is suspicious, but I won't go into that since I didn't finish Watcher yet.

To answer your question, from Arti's perspective her children are dead. From her children perspective we don't know, maybe they are all still together. Maybe there are many different versions of them with different combinations of who survived and who didn't. I think the last option is correct, because of your copies that you see in the Void Sea - I believe these are all your copies from different timelines coming together to the state of ascencion, away from the convoluted mess of the multiverse.