I really grow tired of grown men who weigh 200-250 lbs. insisting that the .243 has "no kick at all" when I see the heads of 90 lb. kids getting snapped back on each shot.
"just get them a heavier rifle then..." some guys will say. Really? You want a 90 lb. kid to heft a 9 lb. rifle to prevent the recoil from snapping their heads back? wow.....
I totally agree with this. This is why I recommend a rifle that fits kids first. It must fit them in weight, and length of pull, and the sights or optics must also work for them. Then, I recommend low-recoil cartridges. I bought Grendel for kids 12-13 y.o. that are under 80 pounds. Full loads with 100 grain bullets at 2800 fps were too much to start with. One worked up to it, but another needs more time with Trailboss loads at ~1400 fps.
I really recommend to do what's best for the kids by starting slow and working up. We know there is some kind of "minimum" to take deer, but a good shot with a light cartridge beats a bad shot with a magnum any day.
Grendel, x39, .243 and similar cartridges all can meet the minimum required to be effective on deer at generous ranges, but not all kids can transition well from rimfire to those cartridges with factory or maximum loads. If you can't handload reduced loads, kids will be best off spending plenty of time with a transitional rifle like .223 before trying to step up to a better deer rifle. If you can provide reduced loads, then anything with the potential to load lighter bullets can work -- .243, 6mm, .257, 6.5 or .277 for examples.
It's harder to reduce the recoil of medium and big bore cartridges. They shoot heavy bullets and without high velocities, the range is limited. I love .357 Magnum and even if you can't handload reduced loads, you can use .38 Special. However, once you work up to hot magnums, the recoil isn't that light anymore and the range is still quite limited. For the same amount of recoil, you could have a rifle that shoots flat out to 250 yards. The bigger bores like 44 Magnum, 450 Bushmaster or whatever still have a lot of recoil for their range. Sure, kids can learn trajectory and bullet drop compensation and range estimation and so on, but they learn those things with range time, not on deer hunts, and by the time they've got those things figured out, they will be able to tell you what they want.