This does not need to be feared. But, how successful you are depends on the adults and kids, and really the kids age. In our case by far the worst age for flying is around 1. They want to move, but they don’t walk very well so they’re on the floor and generally into everything. Airports are DIRTY places.
I’ve checked car seats loads of times. Bring your own bags, though. 2 car seats max per bag (really only an issue once you get to boosters). Mine ride in the airplane seat once they turn two. We’ve used GoGo’s with the kids in the seats and a luggage cart with both car seats attached. If the seats are necessary both of these work. But, they’re both a royal pain going through security and boarding. I have narrow car seats and I couldn’t roll mine down the aisle. Gate checking a double stroller is so much easier. We give a backpack to each adult and that’s it for carry-ons.
In general, I prefer to carry-on less stuff. Its been a long time since I’ve taken a trip short enough to carry everything and if you’re checking anyway lighten your load. I try to avoid all belts and other metal stuff that might have to be removed. I also put all our phones, games, etc together in a clear ziplock bag; you can remove that from your carry on and send it through security as is. Each child must also walk through security alone so send an adult first to catch the stuff and them and use the other adult as a sweeper. Their lovie will have to go through the machine so be prepared.
And, don’t buy drinks before security; they can’t come through. I take trail mix and apple slices for the kids. It takes them a while to eat that. We just don’t worry about meal times; its something to do. Make sure you bring enough food for all ages in case you’re delayed. And, if you have little ones a change of clothing for them and you is probably warranted. I bring a book per child, a new set of matchbox cars, and the electronic games for the older ones. Other than that, hope all goes well!