I put my daughter in a XS ski helmet as soon as she could wear one. If I had it to do over again, I'd find a hockey helmet because they have cages and you can use a shield instead of messing with goggles. Full face BMX helmets may be hard to find and some are just as heavy as MX helmets, be sure to check weights.
I bought her a mx helmet (youth small) on sale, figuring she would grow into it. Next thing I know, she is in a youth large at 5 years old. Must have Dad's melon head. She wears a neck brace with the helmet. Don't worry about restricted movement because if they are riding with you they will fall asleep anyway and the brace helps keep their head from falling over too much.
I also use a sledpal harness and thing they are great. Highly recomend to any parents. If you have to stand for some reason, the kid comes with you and won't be thrown if you hit a sudden bump or something.
My daughter has been riding with me on snowmobiles and atv's since she was one or so - maybe younger. She now rides with me on my motorcycle and wants one of her own. When she shows the motor skills, I'll get her an atv or bike. She has the judgement now. Their is an old phazer II waiting for her when she is ready for a sled.
BTW, I still have a soloman ski helmet XS (black) and Youth S MX helmet (orange) if someone needs one, PM me and we can work something out.