Very cautiously and most likely using some sort of easily manageable weapon (knife or gun if available) to make up the difference in size, experience, bone density, and force generation.
Here’s the thing, as much as some societies have loved equating women with children (and, in some cases, animals) for a very long time, there are quite a few substantial differences between a child and an adult that hamper a child’s ability to fight (even when fighting other children).
It’s not just that children are smaller than their adult counterparts, they aren’t physically mature. It’s not just that their legs are shorter, their bodies are smaller, and their hands are tiny. They’re still growing and any serious damage inflicted on them can significantly hamper that growth (both physically and emotionally).
For example: when I was twelve I broke my leg in a training accident. It was a spiral fracture to the tibia and required surgery to be fixed. The surgeon wanted to install a metal rod in the bone, which was standard procedure for adults at the time, however installing the rod would effectively end any further growth in that bone. While the rest of me would keep growing, that part of my leg would not. This would mean that when I finally achieved full size one of my legs would be several inches shorter than the other. This would have lead to a lifetime of significant difficulty. My parents insisted that wasn’t an option even though the second option required a longer recovery period and that’s why my legs are the same length today.
Softer bones lead to easier breaks. A child and even a young teen cannot generate the same level of muscle mass as an adult, even though their muscles have greater elasticity which means they can develop better flexibility. Serious physical injuries run a greater risk of being crippling or becoming so during the recovery process (because again, their bodies are growing and changing, you can’t rely on their bodies remaining static).
They are less capable of generating force both as children because they are very little (in a way that women are not), very light, and much less capable of making use of the body’s full rotation with hips and shoulders (though they can learn it).
They lack coordination. Children are inherently more physically awkward than adults. It takes them slightly longer to develop balance, coordination, and flexibility. While they do learn quickly, everything that you need to make a fighter happens more slowly. When you train them, what you’re doing is molding clay. Instead of thinking about a fighter who can fight right now, or within a few months of training, it’s important recognize that a child is a long term investment.
You’re building an exceptional fighter both in mind and body, but you won’t have one for ten years.
Children and even young teens (under eighteen with a lower limit of sixteen) lack the necessary emotional maturity to handle combat, especially against an adult. Even when a child can intellectually understand what it is they are doing, the emotional component to comprehend and deal with their experiences is still developing.
Violent action will shape their personality and too much, too fast can lead to long term psychological scarring.
A child cannot transition as quickly into the necessary mentality as an adult, though after years of training they will be capable of the snap shift. Once an adult attacks them, they will be at an extreme disadvantage both physically and emotionally. If they have more training than the attacking adult, this may help them some, but ultimately they will still run the risk of being physically and emotionally overpowered.
How to beat an adult:
Premeditation and planning. The child must intend to harm the adult in question, if they try to fight face to face, they will lose.
Strike in a moment of weakness. If the child’s abuser is a member of the family, the child will have other opportunities to attack their aggressor in a moment of weakness such as while they are sleeping or passed out.
A teen will have a better chance against an adult, but they are up against the adult’s experience and their authority. Both these things can be insurmountable if they try to face them in “honorable combat”. A child has to play to a child’s strengths, they can’t bean an adult on the adult’s terms.
Working together. Kids who have come out of paramilitary training regimens will exhibit a level of extreme discipline (as opposed to the adrenaline junkies the Dauntless breed in Divergent (they essentially make gangs) or the emotionally unstable/anger management plagued Careers from The Hunger Games), while they won’t have the level of emotional maturity to comprehend the effects, they will be more than capable of taking down an adult if they can get the drop on them first and if they are using superior force of arms.
Superior force of arms. Having a weapon like a knife, a gun, a baseball, bat, or a lead pipe will allow a child to take on an adult. So long as they can overcome the presence and authority an adult has over them and their mind.
Raising kids to combat:
It’s important to remember that, as I said above, training kids is an investment to the society’s (or corporations, or military’s) future and because they are, they get handled with a certain amount of care. Kids raised to combat such as the pages and squires who eventually become knights, or the kids who come out of many of the modern day paramilitary programs scattered around the country aren’t child soldiers.
They don’t behave like child soldiers and they don’t get treated like them. Child soldiers are a different psychological animal.
These kids have value. They are cared for. They are respected. They know and understand their place in society. They usually get trained with an insular mindset and are hardened against outsiders.
When you want to write kids who have been trained from birth for war or even just for the arena, you have to stop and consider what they are being trained for and why.
What makes a good soldier?
What makes a good warrior?
Why does this society need to train these kids young?
What is their value?
The more substantial the time investment, the more valuable the children are to the society or military they will be employed in. It’s important to understand that. Unlike child soldiers, kids that are raised for combat aren’t shock troopers. They are specialists and highly skilled operatives. This develops a very different kind of personality and society. And it’s hard to do well, especially if you start by looking at the wrong source material. (The Spartans only work if you look past the brutality and understand what they were trying to create and teach the kids.)
This gets washed over in American society because of the focus on youth and youth being good (high school is the best years of your life bs). The truth is that experience is more valuable than youth, it trumps it. Unless the child starts training early (and even if they do), an adult will be one or two (or twenty) steps ahead of them. If they’ve been in more battles then they’ve seen more, they’ve grown more crafty, they’ve sharpened their cunning, and they’re no longer slaves to their hormones.
Fighting, for them, isn’t new anymore. This is the most important concept to really internalize when you’re writing characters who fight because the aged mentor figure is important and is, even when old and wizened, exceedingly dangerous.
Someone who has survived through enough combat to reach old age is someone who is very good at what they do.
Age and Treachery is a nasty combo, even against Youth and Skill.