Hey, I was you a few years ago! A mama with back pain who took her 3 kids between 4-7 to Europe for three weeks for their first trip over sea. (My partner also suffered from back pain, though less so. ).
Everyone's points here seemed very helpful and valid, and I have to add my two cents from someone who's done it many times since.
We did all carryons, never checked luggage. One- because we didn't drive in Europe and knew we had to do public transportation, and two- we figured if we couldn't carry it, we shouldn't bring it. There's nothing worse than after a long flight having to wait for luggages with cranky, tired, hungry kids. (Off the flight and they were happy to go in search of treat snacks rather than wait for luggage)
We had RS convertible backpacks for the adults, along with a daypack purse for me and a small daypack backpack for partner, and the kids' used backpacks that fit their size. (We went to REI for one kid's backpack, and the other two younger ones used their familiar school backpacks)
Kids carried 2/3 of their stuff in their own packs, and we adults carried 1/3 of each kid (in case they lose their luggage and we didn't notice) we would still have some clothes for them.
I did consider the wheeled backpacks for the kids but very glad I DIDN't because, I learned that:
1- that meant I would constantly have had to keep track of 7 moving parts (3 kids & their wheeled luggages, plus partner) versus 4 moving parts.
2-very importantly, in airport and in train stations or anywhere for that matter, partner and I can hold onto the kids' hands as we lead them through crowds, much easier to do than if they were managing wheeled luggage. (5-7 year-old kids and wheeled luggage in moving escalators, through crowds etc... are more of a hazard to themselves and everyone around them, I've found).
3-most importantly, I confess I wanted my active kids to be... cough, cough... weighed down somewhat so they're slower and easier to manage through airports/train stations. My Mama-bear survival thinking was--what good would it be for me to be heavily weighed down with their luggage and they're free to run off in crowded places?
Did my kids whine and complain and refuse to carry their packs at times? Yes, yes, and yes.
But then, they whined and complained and refused to cooperate daily, (may be even hourly), at home, (it's par for the course with having kids, I think) so my partner and I simply dealt with it as we'd done at home, depending on the situation. (Consistency was our goal, but flexible negotiable for special situations (hunger/tiredness/sickness) and occasional bribery when needed.)
Our kids are now older, and we still do carryons only with backpacks, and darn it if our teens didn't offer to carry 1/3 of their parents' stuff to ease our backs this summer. Hell, yes, I took them up on their offer.