Let's break things down here a bit
Stroller leave it at home and rent one from an outside company once you get there
Carseats strap these to an inexpensive luggage cart (we got ours at
walmart) using bugie cords-make sure every adult knows how to set up and remove this before you go. Both kids or at least the 3 year old can ride buckled in a seat (mine loved it so much at a little over 2 we had issues with trying to climb in the seat before we had the straps out-we used a carseat bag and were able to put the entire seat and cart in the bag tossing DH's heavy winter coat -which would not be needed until we got home and we were being picked up at the airport door by friend's bringing vehicles up so why he packed in along I have no idea-on the seat once it was in the bag. That got tossed a few times before DH could get it and strap it to the top of the seat so a certain small person could sit).
If the 1 year old does not ride in a seat I would use a child carrier. One of my favorites is a ring sling with an unsewn tail. (hint you can take and put it through the handle bar of a stoller tighten so the rings are against the bar and drap over a stroller for small child who needs a nap or even for sun protection. The one I used was a lightweight batik one from taylor made. This same company also makes a mesh style which is great for in the pool or even a shower.
Carry on luggage I would recommend a combination of backpacks and duffle bags. Each adult can put on a backpack and depending if both children are riding in seats put a duffle bag with a longer strap cross body or the duffle bags can be put in the 2nd seat along with one one of the backpacks (bring an extra cord to go aound the top or use the carseat straps to buckle them in if that works. 3 year old would be allowed carry on item so I would recommend a small backpack that can be held on the child's lap. I would suggest that one of the backpacks be the diaperbag for both children including an emergency change of clothes for all adults (in case of a child having an accident all over an adult) and a small colapsable cloth shopping bag (I have one that folds up to the size of my palm) so when a kid seat cover change is needed they don't have to take the entire bag-just grab that bag toss in the items needed and head to that airline restroom. I would also advise having them practice changing the kids while the kids are standing up especially the 3 year old as there might not be space for them to lay down especially if they don't have access to a handicapped restroom on the plane.
There is a blog written that has a bunch of stuff that you would ever want to know about flying with children by a former flight attendant and expat american who has personal experience flying between europe and the us several times a year I can either post a link here or can send it to you via private message whichever you would perfer. A lot of the tips on here helped me when we were traveling with our kids.
DO NOT CHECK THE CARSEATS The chances of them getting tossed around are extremely high and carseats have been damaged this way. When they are gate checked they are loaded onto the plane last and removed last along with other gate check items such as strollers and wheelchairs. These are treated more gently than the general checked baggage. A damaged carseat is not going to protect in case of an accident and chances are your auto insurance can toss out any claims if those seats were checked as baggage on an airplane just like they can if you have a minor fender bender and because the seats don't look damaged don't replace the seats and then get in another accident and the child in the seat is injured. I know my insurance company told me that is the case if we choose to check a carseat as baggage as we had planned to do with our then 5 year old's seat (we ended up geting one from the car rental company and I have reports that it was brand new from the members of our party who went to pick up the vehicle on the day we needed it to go off property-helped that the arrangements were made with the manager of the company we went with at our local office and we know the manager personally so we got brand new out of the box seats for all the kids were needed ones for as they took them out when the vehicles were picked up and installed while they waited)
Another option since it sounds like your going down ahead is to purchase a couple of inexpensive seats such as the cosco scenera which runs around $45 use those and then pack them back in the boxes they came in from the store and either ship them home with a shipping company (cheaper option if someone is taking a later flight and can take the seats after the kids are dropped off at the airport to fly home) or check them as baggage. In the boxes they would be protected as the companies have designed them to be shipped safely in them with all the packaging.
Advantage of this is they would have extra seats if they were ever need them. I know we were thankful to have a couple of inexpensive seats when we were rearended by a vehicle going 65 in a 35 while stopped (3rd vehicle back) at a red light. It was nice to be able to call my inlaws have them come get a key to the location of the seats go get them and bring them to us-we were in a heavy enough vehicle being in a half ton truck that we had very little damage-the two little compacts in front of us were significantly damaged-we were hit by a jeep/suburban type vehicle that had enough front end damage it was not drivable-didn't help them to have hit our hitch
