frizzller made a good point, that the .243 hold lighter bullets and less powder than the .308, making it lighter kicking in the same weight rifle. However, the funny thing about it is, although it may not be noticeable, if you loaded a .243 and a .308 to the same muzzle energy in the same rifle, the .243 would kick harder, because it would be trying to push the same force out of a smaller tube, therefore the internal pressure would be much higher in the .243 load than the .308 load, which would give a harder recoil, that's why they load down the .243, to stay under SAAMI standard internal pressures.

ANYWAY.....In most rifles, I don't think that the .308 would be unmanageable for someone 10 yoa. In most factory loads, I'd imagine that most 12ga 2.75" pump action shotguns kick harder than the .308. I gave my fiance her .308 when she turned 16, she was 5'5", 115# at the time, and was, of course, a girl!!, and she never complained about it kicking too hard. Her brother started shooting a .270 when he was 12, and he was around 105# at the time. This Thanksgiving, my cousin came over with his high powers, and all the kids (boys and girls) wanted to watch while we shot, eventually it came that we offered to let them shoot (with forend on a haybale since they weren't strong enough to level them), an array from .22-250 to .308 as options. The oldest of them was 7yoa, youngest was 4. Most of them shot the .308, and not one complained about it hurting them. How light built are your sons that recoil that didn't hurt a 30# 6 year old is going to hurt them?

Give the kid a pair of muffs so the sound doesn't scare him. Even if felt recoil isn't much, if the sound scares you, you'll think it hurt like heck!

If it were my son and he couldn't handle the recoil of a .30-06, I'd buy a .308, put a good recoil pad on the buttstock, and buy him a padded shirt to benchrest with and not worry about the hunt, because he's not going to feel it then anyway. Maybe looking into an A-bolt with BOSS system, reduces recoil, but make sure they wear their ear muffs!

Other suggestions would be a .30-30, 6.5x55 swede, 7mm mauser (7x57mm), 7mm-08 or go ahead with the .243. Any one of these I've mentioned here are probably going to give you a bit better guarantee than the .243, but with less recoil than the .308

Screw the 10 ring, keep them in the zero!!!