As with all of my equipment, I looked to the Japanese for examples of what they used. Early on (2012) by chance I noticed in the background of a Dr. Ishigaki fly tying an interesting looking case that he used to carry his rods. During my visit to Japan in 2013, I shopped for equipment at Sansui and Joshuya and purchased a GETT rod case that I use to this day.
I use the cloth rod sleeves that come with most rods when I store my rods inside the zippered case. They prevent the rods from getting "road wear" from vibration and or movement during transit. The cases are easy to open full length and you can keep the sleeves inside while you are using the rods.
For air travel, I use a luggage tag just in case I get separated from my rods. On one flight back to the states from Narita, Japan, I as asked to check my rods as luggage (no extra charge) which I strongly objected to but lost my battle. The rods rode in the luggage hold with absolutely no damage to them. This was the deciding factor in purchasing more cases to hold the varying lengths of nested rods.
The zippered rod cases from Joshuya branded GETT will not be easy to obtain. They are not readily available by mail order. You may find other cases from brands like DAIWA that may be easier to purchase. Each of my cases where less than $20 USD and have been protecting my rods from all the damages of travel by car or airline.
I highly recommend using them for travel to your favorite stream or halfway around the world.