If you have several nests put a gate in front of the one shes in. Left on their own the other birds will be pushing in to lay and may push her out completely. When that happens here the broody always seems to adopt another nest, which means she starts brooding more eggs and lets the first clutch cool. You can lose a lot of eggs this way. To remedy the matter I put a hinged gate on the separator wall between two nest boxes which means I can have a broody in either of two nests protected from intrusion. I keep water and a bit of food in there for her, and phyically bring her out to poo once a day. If you don't want to take one of your nest boxes out of action for three weeks, go with a pet carrier on the ground.
As far as moving a mother without hurting chicks, just reach carefully under her breast bone and lift slowly. Move the mother to the new spot first, with one chick to keep her from panicking. When you move the rest over, place them in front of her, not under. They will crawl under on their own and inadvertant chick squishing will be avoided.