|Moving the hen and clutch IS tricky and you have to be quick.
Have a dog kennel (or similar sized cage) ready inside an outbuilding,
not necessarily your chicken coop. The dog kennel is nice because offers
more privacy like she prefers. I put mine in a more secluded section of
the barn rather than directly in the noisy hen house. Have straw in the
kennel, and water hanging on the door where you can fill it easily for
the hen. (I actually have parrot food and water dishes hanging on the cage
door and use a plant watering can to fill it through the fencing rather
than to put my hand in there at first.)
You will need help with this move, preferably a strong man dressed in thick sleeves and gloves.My husband does this part for me and I assist. He sneaks up from behind her while I am in front and she has her eyes on me. She will put up a fight when on that nest, so do be careful. Her bite is horrible!!! My husband quietly and quickly bends over and grabs her over her wings, face AWAY from him, and holds her out away from him as he quickly moves to the kennel already in place as I quickly put each egg in egg cartons and rush to get ahead of him. I put the eggs in a clutch as she had them in a prepared NEST in the back of that kennel. He puts her in and we quickly shut the door. Believe me when I tell you this is done FAST.We have carried a hen from as far away as nearly 1/4 mile. We have even done this when eggs when to our surprise I scooped up hatched keets and eggs in the process of hatching! (I actually carried baby keets in the egg carton too - very carefully of course because the lid would not close!)
Each time we have done this it did work sucessfully, and the hen did complete her hatch and the raised her keets in the kennel until we let her loose and took over ourselves. Each time we allowed a hen in the field to hatch her own, we either lost 100 % of the keets to owls or other predators when the new mother would leave them to go eat a few bugs, or lost the hen before she was able to hatch her eggs. For these reasons, we ALWAYS take eggs from nests, and when we find a broody hen, we sometimes use the kennel method.
If you are in a safer area than I, and feel no predators will eat your hen for the 26-28 days it will take her to hatch her eggs, by all means allow her to do this on her own. If you plan to move a hen and her clutch, do not disturb her nest if you find her off of it. She will not go broody until there is a clutch of 25-30 eggs in it! She will not go broody just because you stick her in a cage with eggs. This method only works for a determined hen who has been found after being broody for several days - weeks.