I would much rather wait to rebuild on Siptah and deploy my thralls there.
Thralls Cobras and Cats don’t have a Decay before being placed so you can just keep them in a Box until you need/want to place them. Other Pets/Mounts you will want to keep in a Preservation box with Ice to disable their Decay. Doing this the only decay you have to worry about the Decay timers for your base components. If you don’t have a base and are keeping them on your character the Pets and mounts have a decay time of 24 hours for Babies and Horses then looks like around 170 hours for Camels and adult Pets. I believe the Apes, Yetis and Giants don’t have a decay on their packaged forms either.
Eggs take 24 hours hours to hatch before the babies are available then the babies have the 24 hour timer start. So you have that time to get them into an Animal Pen or Preservation Box, Snake eggs being the exception since the Cobra babies don’t have a decay timer to worry about after the egg hatches.
from what I observed, the timer does not apply to characters in the cloud. What happens if you transfer a character but don’t deploy a follower that has pups in his inventory and is not deployed, I don’t know