People do everything. There is no one right answer for everyone since we have different circumstances, different goals, different management techniques, and our chickens have different personalities.
Some people move a broody and make sure she will remain broody before they give her eggs. Some people move a broody and her eggs before they hatch. Some wait until after the eggs hatch but before she takes them off the nest. Some wait until the broody brings her chicks off the nest and abandons the unhatched eggs. Some do not move her or isolate her at all.
I would never move a broody after she has started setting on the eggs from worrying that she could break from being broody. That's just me. I wait until the broody has completed the hatch and brough ther chicks off the nest. Then I put them in a predator proof isolated pen for a couple of days so the chicks can learn to eat and drink without the interference of the other adult chickens. After a couple of days there, I let them loose and allow the hen to raise them with the flock.
There are different reasons I let Mama raise them with the flock. It is easier and more convenient for me since Mama does all the work, but the main reason is that Mama will take care of integration. She won't take care of pecking order stuff, they have to do that themselves when they mature enough to handle it, but basic integration is not a problem. To me, that is very important.
One big thing that allows me to do it this way is that I have lots of room. Mama has plenty of room to keep her chicks away from the flock most of the time so she avoids certain potential conflicts, yet they also spend time with the flock. I've also had broodies that will protect their chicks. Almost all will, but every now and then, you get one that does not do a good job of that. There are risks any way you try it. When you are dealing with living animals with their own personality, you never really know what will happen. But for my circumstances, letting Mama raise them with the flock has been very successful. For a few others, it hs been really bad.
Good luck with yours.
