Everybody seems to have their favorite method for this and the "best" approach will depend a lot on what carseats are involved, how old the kids are, whether you have another adult with you, how far you have to walk when changing planes (if applicable), and so on.
For our family, we usually just pile the carseat(s) in an empty stroller, leaving the kids to walk or be carried/babyworn, depending on their ages.
Some carseats like the Radian can also be worn as a backpack with padded shoulder straps, leaving your hands free. I've sometimes worn the Radian and then carried a Britax.
I've never used any of the various wheeled carts, but they do look convenient and some folks swear by them. I'm seeing them more and more often at airports, too. You mentioned that the one you tried didn't fit, though.