People do it a lot of different ways. Some of how you do it will depent in the personality of your chickens and how much space you have.
By personality I mean that some flocks have chickens (usually hens) in them that will search out and try to destroy young chicks. Most flocks don't have those, but some do. Most broodies will defend their chicks. Most, but not all. Some roosters protect all members of their flock, including baby chicks, even from other chickens. Some don't. Most of the time, the other chickens will ignore the babies unless they get in the older chickens personal space. Then they will probably peck the chick. If the peck is just a reminder that they really should not be that far away from Mommy, the broody pretty much ignores the peck. If it is a vicious, I'm going to kill you, type peck, the broody will probably protect her chick. With most flocks it is not a problem, but sometimes it is.
If they have room, most broodies keep their chicks on the outskirts of the flock. This avoids conflict with other chickens. Sometimes they do OK in fairly tight spaces, but the more space you have, the less likely their is to be a conflict.
I don't know how tight your overall space is. Assuming a reasonable amount of space, what I would suggest is that you keep Mama and the babies in their enclosure for maybe two days after she brings them off the nest. Just long enough for them to get used to eating and drinking without any interference from the older chickens. Then let Mama bring them out to meet the flock. I will not guarantee that this will work, but it is very close to the way I do it.
Editted to add: I assume your cage has the wire tight enough that the chicks cannot get through it. They are in real danger if they can get through the wire and mix with the rest of the flock if Mama is locked up where she cannot protect them.
