Coupla questions? What is a small kid? My grandsons age 8-13 all shoot my .308s just fine. After explaining recoil and how rifles work, they revel in the ability to control a full size rifle.
Second, the .243 is a wonderful youth cartridge as-is. I've settled on one load for mine, the 100 grain Hornady bullet and Reloder 22 powder. It's a full-house load, but it recoils much less than my .308 loads.
If the young-un is able to hold the rifle to his shoulder without assistance, he should be able to handle a .243 with factory ammo. If not, then maybe he should wait a year or two.