From Science Daily, March 2011 regarding a study done by The Center for Injury Research:
During the 19-year study period, an average of 8,700 boxing injuries
were treated in United States emergency departments each year, and
approximately 2,500 of those injuries were to children and adolescents
6 to 17 years of age. The number of boxing injuries each year
increased 211 percent during the study, climbing from 5,361 injuries
in 1990 to nearly 17,000 injuries in 2008.
The most frequent injury was a fracture (28%) and the most common injury site was the hand (33%).
Injuries from boxing bags apparently happen when the bag is too heavy (there are child-size bags, but they are still about 35-40 lbs, so your child should weigh at least twice that as a beginner), when the boxer has bad technique (the bag swings instead of "pops" because the boxer is pushing instead of jabbing), when the wrist isn't kept straight, when balance is off, when hands are not wrapped and gloved... There are reports of hand, wrist, elbow, and shoulder injuries as well as injuries from being batted by the bag as it swings. Bottom line, this probably should not be attempted without proper training. I have found gyms that will start training as young as 8, but they generally suggest that children younger start with a different type of martial art in order to develop some punching skills before hitting a bag (or person).