Sometimes with fast-growing kids, it's tough. It seems like every time my 6-yo gets on her bike, the seat or bars need to be adjusted. Usually we're still on our own street, so it's easy to swing back in to the garage for a tweak.
Since kid's bikes often have hex-head bolts on these, look for a set of 'ratcheting box-end wrenches' These will make those jobs much quicker. You may find that seatpost and handlebar stems on the kid's bikes are only one or two sizes, so you just need to bring those along, not the rest of the set.

If any of your bikes need to have on-the-road repairs or constant adjustments to finish a 5-10 mile ride, you need to do a more thorough job of going over them, or have a qualified tech do it, as this is a sign of a bike in bad repair.

I do carry a spare tube, and patch kit, on my bikes, since they do see longer rides, but the patch and pump work on any of the bikes. Most of the bolts on a modern (non-Walmart) bike are going to be allen-head sockets, so one of the 'swiss-army-knife' style multi-tools is sufficient.

I do have a bolt-on rear wheel on my vintage (1976) ten-speed, so I carry a small adjustable wrench to assist with removing the wheel for flat tire repair, but that bike has been re-built several times, and over a pair of 100-mile rides, it didn't need anything more than a trim adjustment on the front dr.

I used to do a lot of back-woods MTB trails, and you had to carry enough tools to get your bike rideable after a crash, or be prepared to carry your bike back to town.