What you can do with a doe depends largely on the doe, and your relationship with her. I have had does that were fine with me messing about in their nest boxes, but anyone else had better watch out!
Your doe may be fine with being moved - most of the rabbits I have had would not have a problem with it. However, if your doe is an extremely nervous sort, a move might not be a good idea. I have had the rare doe that abandoned or even savaged her litter when she got upset, so I know that it can happen, though the vast majority of my experience has been that it wasn't an issue. You didn't say how old the litter is? If this doe is comfortable with you, it would probably be fine, but it also might be a good idea to wait until the doe has gotten used to this "motherhood" thing before changing things on her.