I once had a "screaming" baby in first. A trip to the Children's Hospital emergency room (after the flight) advised us it was reflux but it was a true flight from hades. As I did not want to upset my fellow passengers any more than they had been (and I do know what do with flying babies/children - this was a new one - I tried everything and FAILED), after the seatbelt sign went off - I went to the front galley/jump seat area and did everything possible to calm her. I was not that successful but she did quiet some and at least we were away from the pax, the FA did all she could to help (because I'm sure she saw the death stares from the other F pax) and we endured the flight. I apologize to my fellow passengers because I felt horrible for them and if there anything more I could do, I would have.

She's 3 1/2 now and gets very upset when she hears a baby crying. I tell her she used to cry and that the baby's ear hurts and that it's important to drink at takeoff and landing.

No, an FA can not/should not move PAX back for this. A parent should do everything in their power to control the baby, soothe the baby, whatever. As for the 8 year old jumping around like a kangaroo - absolutely no excuse. The mother just didn't do her job in parenting.