Move them at night. Put the chicks in a box, get mom in the new nest box, then start putting babies under her. But, as others said, wait til they all hatch. Mom usually waits three days then takes the chicks out for their taste of water and food. Hay is ok for now, but, change it to shavings before two weeks. In my experience raising young chicks, they always end up eating the hay and one or two always end up crop bound. They usually start this by week three. Mom may even eat it if penned up. So, best to change it out as soon as you can.
Oh, and if they are going to be exposed to grass or even shavings, get the baby chick grit for them. You can start giving that to them when they are two weeks old. Just a little at a time. Sometimes they really gobble that up.