It's fine to move them! Making a nest out of a box and towels might be appreciated by the mother. She won't hurt the kittens. At most she'll get upset at you for handling them, depending on how much she trusts you. Just make sure you do all your handling of them in front of her so she can keep an eye on where they are.
So, this is quite soon for mom to have another litter and it's hard on her body. Can you give her kmr to drink every day? She needs the calcium. She also needs to eat as much dry and wet kitten food as she wants while she's nursing.
Can you keep her inside and away from intact males?
Be sure to get her spayed this time when the kittens are 6-8 weeks old. You really need to make sure she has an appointment set up and does not get outside until then. She'll be happier and healthier and won't run off or have more kittens. [emoji]128522[/emoji]
It's a good sign that the babies are nursing well. Mom cat will likely take very good care of them.
If your mother feels they are okay where they are, then just keep an eye on them and feed mama cat lots of wet food and dry kitten chow. She needs it while nursing.
Post a picture for us when you can! [emoji]128522[/emoji]