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I wouldn't advise moving her or the pups till they are at least 3 weeks. She may well cull them if you did due to the stress.
I'd suggest getting some cardboard and using cable ties to attach it to the bars right round so she can't get out. Place the whole thing in a big box and hope she stays in there.
You could alternatively attach chicken wire all round for now.
When you move them you need to move all he dirty litter as well as the intact nest of you can. I would normally putthemin for playtime at around 18 days old and then back in the cage again. Put mum and pups in and see how it goes. If she settles with them you could leave them but if she looks very stressed then she needs to go back in the old cage.
I hope she manages OK. If she can get out then so can the pups so you must secure the cage.