When Moo was pregnant I put her into storage
Only kidding! I put her in her storage bin a week before she had her babies, that way she got used to it happily. She had her babies in the storage tub and at 2.5 weeks I moved the whole family into Moo's original cage (base level + 3 levels so...)
It depends though, I knew Moo hated being in a tank rather than a wire cage... but she didn't mind babies being moved at all, especially as I just placed her whole nest into the new cage lol
but she loved the freedom to escape!, mind you, it didn't last long!