So here is one more approach to the problem of carrying kites on airlines. The airline in question was SouthWest, which has arguably the most liberal policies around, but the only real advantage we took of that was that up to two checked bags are free. The keys to our solution are that the checked baggage rules (normal, not golf clubs) are a total dimension of 62 inches (that's length + width + depth) or less, and that almost all our framed kites break down to less than 36 inches in length.
We took a 10 inch (nominal - more about that later) diameter tubular concrete form. These are made of a heavy duty cardboard (about 1/4 inch thick) and sold in home centers in convenient 48 inch lengths for a few dollars apiece.
I cut the length down (on a table saw, very carefully!) to 38 inches with the idea that 38 + 10 + 10 is noticeably less than 62, even when enclosed in a bag.
My wife observed that the 9 inch cakes pans we had just discovered in our basement fit nicely inside the tube, with some duct tape just to hold the bottom one in place:

The second cake pan is shown sitting on the tube, which is bottom up to show the duct tape holding the first one. Then we filled the tube with kites (10, as I recall):

Put the other cake pan in over the kites:

We then took an army style duffel bag, opened up its seams, removed about 12 or 14 inches from the circumference of the bag and used the material to make the bag about 14 inches longer.
The tube fits in the bag, with some room at the top for soft stuff (a parafoil), and it all closes:

The tube is stiff enough, especially when fully stuffed, to protect the kite sticks. The cake pans both close the ends and add stiffness to the cylinder ends. If you're still concerned about protection, the nominal diameter of the tubes is sufficiently variable that it's easy to buy two, one of which just fits inside the other. In fact, they're stacked that way in both my local home centers.
Anyway, SouthWest accepted it without question or comment and there were no dents, much less kite damage, when it came off the plane on both halves of the round trip.
Larry
Edit by Mike - Fixed your Image tagsThanks - Larry