I’ve successfully moved 2 of 2 broodies now…I’m not sure if I’m extremely lucky or if I have an unknown something going for my system by luck, but here is my method, most of it gleaned from others here on BYC, for what is worth:
I set up a small 30” dog crate partially wrapped with fleece (to give the broody a bit of privacy, and also to keep new chicks from escaping. I separated the crate into two parts with a 1x3 board that just happened to fit, mostly to save some on shaving usage but also to give the broody a nest fairly similar in size to our nest boxes.
In this picture you can see the inside of the broody crate…the purple is a towel I will talk about in a minute, the multi colored fabric is the fleece, and there is one egg in the nest side of the crate. I give them an infertile egg or two the first two days, and once they have been setting a few days, I give them the fertile eggs.
When I have the crate set up, I put it in the corner of the coop, opposite the nest boxes…mostly because that spot is out of the way and under the poop board, but it being across the coop from the nest boxes has also helped in training her to her new nest.
The morning I’m moving my broody, I pick her up out of her chosen nest and set her on the floor…she dazedly shakes, and runs off to do her business. When she comes back, I grab her as she is entering the old nest, shove her in the dog crate, close the door and cover with the towel the part of the crate not covered by fleece. I then retreat to the house and check on her via camera in the crate. For both hens, once I covered the crate with a towel, they were setting again within 3 minutes (we had a failed attempt without the towel for my first broody and she was still losing her ever-loving mind after 15 mins, so I think the quick cover with the towel might be key).
Broody #2 who originally gave the new nest 0 out of 10 stars…when she first set down after being unceremoniously shoved in (because I had a shrieking flapping pancake trying to get away from me as soon as I grabbed her

), she didn’t even set on the nest side:
However, within a few hours she moved to the nest side:
Leave her closed in and covered all day and through the night. Next morning, when I let the chickens out, I also open up her crate and she storms off to do her business. While she is out of the coop, I pull the nighttime nest box curtains closed, close the pop door and wait by the human door, keeping other hens out of the coop (to make my life easier, keeping nosy - and higher pecking order - hens from foiling my plans), and once she goes in the coop, I walk in behind her and close the door, then go stand in front of the nest boxes (because she’s still mad at me from the catching-and-shoving-in-a-new-nest incident from yesterday

and wanted nothing to do with me). Generally she grabs a few more bites of feed, then turns to go to her old nest — only to see the curtains closed and me there. She gives the closed nest boxes another few stares then heads to the new nest after deciding she didn’t want to tangle with me over a closed nest box. I close her in and partially cover the front of the crate with the towel again, but leave enough open that she can tell it’s daylight out.
Here is picture of our coop, with nest box curtains closed and broody cage in lower left (first few mornings I stand in the upper right corner of this picture):
Next day, same thing - let her out, close nest box curtains and wait for her to return. This day she usually pretty quickly goes to the new nest. Today I don’t cover the front of the crate at all, so she can see the flock and they can see her. Next day, same thing, only I don’t stand in the coop, but I do still close the curtains. Fifth day, I don’t close the curtains but watch to make sure she goes in new nest. If she is an alpha hen, I stop closing her in at this point. The hen I have sitting now is lower on the totem pole, so I will probably continue to close her crate every day just to keep anyone from trying to bully her off the nest.