What people often don't really seem to understand is that it is the location of the nest and not the actual eggs themselves that the broody hen homes into. Only towards the last few days of incubation when she can feel and hear them and starts talking to them, does she form a bond with the actual eggs (or rather the contents thereof). This is why it is possible to give broody hens the eggs of other hens or indeed other poultry to hatch. So, if you move her nest, the next day when shegoes for her broody break, she will try to return to her old nest site. If they are fastened into their own broody enclosure they will usually give up and go to the new nest and settle on that and after a day or two, they will be homed into that new nest location. If you move their nest, but they still have access to their old nest location, they will more often than not return to it as that site is already programmed into their system, even if there are no eggs at the old nest site.
If you keep her in a communal nest box, you need to mark her eggs and remove any others that are added by other hens on a daily basis. But it can happen that she goes for a broody break and comes back to find another hen on her nest and then she may climb into an adjoining nest if there are eggs in it and her eggs can then get cold, so there are pros and cons to moving a broody. It is not really a natural situation for hens to hatch chicks in communal boxes but if they are penned, they usually don't have any other option... no hiding places for them to make a nest of their own. Whereas, if they are free ranged, they will often seek out a quiet hidey hole to make a nest and lay their eggs away from other hens. Because the location is notably different from the communal nesting boxes, the hen can't make a mistake and get back onto the wrong nest.
Personally, I like to keep my broodies within the flock but I have a cupboard in the hen house that I fasten her in with holes in the door for ventilation and an old drawer for a nest inside. I put food and water in there, but let her out once a day for a broody break. Broodies go into a trance like state when they are incubating eggs and therefore a dark, quiet place where they are not disturbed is best. Allowing her out with the flock once a day ensures that she remains part of the flock, but I can supervise her broody break and ensure that she goes back to the right nest and then fasten her back in and know that she is safe on her nest for the rest of the day. They quickly realise that when the door opens, it's time to get off and take a break. For me this is the best of both worlds. Of course, if like me you free range and a broody has a secret nest rather than a communal one, and I can make it safe for her (or can't get to it...I've had a couple that were really sneaky) I just leave them to it.
Enclosed pens and communal nests are usually the cause of eggs getting rolled out, cracked or accidentally abandoned. The broody often gets blamed, but in my opinion, it is the close confinement with other chickens which is the root cause of the problem and this is when a separate broody pen can be helpful, so it depends a lot on your set up.
I agree that having portable nests is useful for this, so that the hen, nest and eggs can be moved altogether and if done at night, the broody will hardly notice, but that can work against you the next day if she still has access to the old nest site, as she will have no idea that her nest has been moved...ie she doesn't pay much attention when she gets off the nest for her broody break as she is just interested in getting a bite to eat and a poop and maybe a dust bath in a hurry and then find her way back to her nest on autopilot when she is ready.
Hope that makes sense.
Regards
Barbara