I've talked about this elsewhere, but I would very strongly recommend going small.
I started my oldest son with a 7mm-08 and reduced 120 grain loads. I regret this. He had lots of rimfire and .223 AR time before going to that rifle. It was still a lot for him at 10, and we've spent the last few years training out bad habits he developed from shooting a rifle with too much recoil too early. My daughter's 10 now, and killed her first deer this past year with 77TMK from the AR. She is way ahead of where her brother was at her age in terms of shooting fundamentals.
He's got a few friends shooting 6.5CM's, mostly braked, but that's going to be similar recoil to his 7mm-08 (not including the difference of a brake). I think a brake on a kid's gun is a bad idea, too easy for them to forget ear pro in the moment.
We have 4 more coming up behind her, so I've decided to get a "first deer" rifle that each will use and then pass down when the next kid turns 10. My local shop has a couple of Howa Mini's in 6mm ARC on the way (long overdue now), and one of them has my name on it. I should probably have done a .223 but just can't leave well enough alone and do the sensible thing. Hopefully it will be here in time for her to use it for one year this fall before her brother turns 10 next year.
I will never put a 10 year old behind a 7mm_08, .308, .270, etc again. Even for elk, it's just not needed if you run the right bullet. That 6mm 108 is, from all I've seen, plenty. And we have all seen the 77TMK thread.