52 years ago my first deer season my dad had picked up this old single shot Ivers Johnson 410 to be my first hunting firearm. Stamped on the barrel was 410ga. 2 1/2 inch shell 44 lug in one. I shot my first deer with a 410 slug my first year which today I think of as pure luck. That is the only deer I shot with 410 slugs as my dad used to get 44 ammo after that year. Don't know if all 410's will handle 44 ammo?

We it me who had a son or daughter of age to hunt deer I would use the 20 ga. Of course you need some range time, so figure out with in a couple shots if the recoil is a factor if so you have choices you can make at that point intellectly. My wife is recoil sensitive so she has a shooting vest she bought that has pockets in the shoulder area where the gun stock fits that hold gel packs. you can buy them in a few different thickness. We also installed kick ese but pads on her guns. She now had no problems shooting her shot guns even the 12ga.
My son started with a 20ga at 10 and was fine with it with just a good butt pad.
Also helps to spread out the range time, rather than do 5 shots as fast as you can reload the firearm spread the 5 shots out to a half hour. gives the youngster time to recover from a shot and also gives you time to talk about the shot (use a spotting scope or good binocs).

Al