I'll give an answer from a completely different point of view. A child who undergoes the stresses of joint locks are subject to the manifestations of injury as described in Bankuei's answer, I have no doubt. But the injury to a child is not relegated to the child, and for different reasons:
Children who engage in joint locks should be considered from both nage/tori point of view, as well as uke's point of view.
As nage, inexperienced children are at more risk of hurting (a minor) uke, because they are not experienced in the technique as well as recognizing when to stop. I often see youngsters perform what they think they see, and overexaggerate the technique; they do it too fast, and don't recognize the signs given by uke to stop. Either they don't recognize the sign at all, or they recognize it too late. For this reason, I see many a martial art school - not just aikido, but other styles as well - have very young and/or inexperienced kids do many of the techniques with tegatana - an open hand. Without the grip, there is virtually no chance of injury, since uke can slide out. Of course, this changes the dynamics of the technique, often to the point it is not useful or practical. But the exercise for these kids at this level is less about practicality, and more about the shape of the technique, an issue to deal with which is outside the scope of this question.
On uke's side, children sometimes cannot exhibit pain: either they don't know how, or they don't react in time. I have seen first hand an instructor bring a child into sankyo, and (accidentally) not realize the child bent over with tears streaming out of his eyes because he was too scared, too startled, or inexperienced to indicate he was in pain. A child being held in kotegaishi can be more easily seen in pain from a facial grimmace than one who is in sankyo, due to the fact that they are more or less facing each other.
Well, these aren't really medical considerations or studies, they are simply observations lending to the idea that children engaging in join locks and wrist bends should be watched with particular care.
However, there is more. When a child is injured - even if not noticeably so - they can have injuries show up later in an x-ray for an unrelated injury. For example, a child sustains a minor spiral metaphyseal fracture because of a joint lock, sankyo, or kotegaishi in class last week might experience minor pain which might not even be reported, or might be mistaken from injury sustained from football or hockey that same afternoon. Then this week, he falls off his bike (whether or not that fall is related to his fracture), and gets an x-ray which shows the other underlying injury. Bike falls, hockey, and football don't usually exhibit spiral fractures, while child abuse often does.
Now the parents have a lot of explaining to do - sometimes to child protection services.
Yes: an instructor who is not careful with his minor students can wrap his student's parents up in a child protective services investigation.
Here are some sources which may be of interest which describe fractures relating to child abuse. Note they are somewhat graphic and unsettling to those not experienced with the subject, caveat lector:
Diagnostic Imaging in Child Abuse:
Non Accidental Trauma
Fractures Caused by Child Abuse
Does this happen often? No, not at all, or else the Surgeon General would be calling for a war on Aikido. But this subject comes up from time to time as a defense when a case goes to a formal investigation and/or court. Parents who are innocent and are accused of abuse in these cases must consider all of the activities their children have done which might have caused the injury. It is not uncommon for horseplay among peers, sports, and martial arts to be blamed.
So, hopefully this isn't outside the scope or spirit of your question. But the subject should cause people to think hard about the training their children get: the injuries their children can receive, the injuries their children can cause, and the (legal) injuries an injured child's parents can receive.
Having said that, there are diseases classified for repetitive use injuries in children, seen here:
Overuse Injuries in Children
Well, you DID ask for studies. In this article, the recommendations and articles are from none other than the American Academy of Orthopedic Surgeons, hardly a group to challenge their recommendations for sources.
I would venture that the US National Library of Medicine and National Institutes for Health falls under a similar category, although, they do source their articles:
The child and adolescent athlete: a review of three potentially serious injuries
Peri-epiphyseal and Overuse Injuries in Adolescent Athletes
As in the AAOS and NIH, the concern here is harm in growth plates and growth spurt. In both articles, the issue is of repetitive action, not the legal issues as mentioned before.
I guess I'm throwing in this one for no other reason than to show that, while the content of the article has nothing to do with Aikido or joint locks, the implication is that with proper and repetitive training, injuries can actually be reduced:
Nine year longitudinal retrospective study of Taekwondo injuries