I hope you won't mind if I butt in here....
If she feels safe in the coop nest box then she at least is confident that she can defend her nest there. They are not often wrong ime.
Coop to coop and house nest box to coop I always do in the daytime and I make sure the hen sees me move her eggs. Most get off the new egg site and try to return to the original site. I pick them up and place them back on the eggs in the new site. I think a hen called Fray holds the stubbornness record. It took about a dozen manual relocations before she settled at the new site.
I move hens that sit at outside nest sites at night for obvious reasons.
You need to mark the fertile eggs and the eggs she laid. It is quite probable that other hens will try to contribute to the clutch. Senior hens tend not to put up with this; juniors are more likely to just sit there and guard their eggs and let another lay.
It is worth making a wire mesh screen to cover her chosen nest box for the night. It saves you worrying about what goes on in the mornings before you let them out. Ime quite a lot can go on in the early mornings.
You can force a hen to sit in most locations and I read a lot of this on BYC. I'm completely against it. If a hen abandons her nest then there will probably be another time. I never force a hen to sit. I only insist on location changes. After that it's up to her.
Provide water at the nest site Bob, but not food. You need to watch that she gets off the nest every day to eat, socialize and most importantly, dust bath. Do not under any circumstances feed her at the nest until the chicks start hatching!
I would not move her once the chicks hatch unless you see one of the other hens attack the chicks. If they attack mum (providing it's not serious) that isn't unusual. It generally stops. One of the great advantages of having a hen hatch in the flocks coop is the chicks hear the other hens while they are still in their shells and once hatched, can judge from their mothers reaction who to avoid early on. It also means mum and chicks get access to where they will be expected to live at an early age. Even in a coop and run situation this has tremendous advantages for the chicks development.
