My foster Sugartoes started moving her kittens at about 2 weeks (I think), and would only move part of them. I moved them all into a large cat carrier with the door blocked open. This way it felt like a safe area to her, yet it was easy to keep clean and see the kits.
A cage is ideal, because pretty soon those babies are going to be walking (and pooping) all over! They get the hang of the litterbox pretty quick, but it is still nice to be able to contain them! LOL! As as foster Mom, I have a cage that is 2 dog cages wired on top of each other, with shelves added. So I could leave the top door open for the Mommas to come and go, and keep the bottom door closed unless I wanted the babies out wandering. Of course, before long, they can jump and climb, and get out the top of the cage pretty quickly, but it did help for a while. (I always left a chair beside the cage for kitties to jump out the top opening so they don't get hurt, and so Momma can get back inside easily.)