I move all of mine to a cage and then give them their eggs. I have had one or two hens that will NOT be move from the spot they have chosen to be broody, but most hens move very well, even though I have to carry them over a 1000 feet from were their pen is to the cages I use.
I have let my broodys sit in the coops before, but I have always had trouble with the eggs rolling out, or breaking, or other chickens getting in the way. If you keep your hens in cages, you have to make shore they have a place to dust though. (I keep a small bowl filled with dirt handy for that reason)