I transport mine in custom made show box transporters, designed to be dark and calming, small as to restrict movement with smooth sides so it doesn't mess up their feathers, it must have a slide opening preferably in the top, no places for food or water as you don't need this and it makes a mess. Try and make the boxes where they hold no more than 4 banty or 2 LF holes for easier carrying and loading, plus when cooping in, line them with plenty of clean large chunk shavings. If you notice almost all show folks use these type show boxes, mainly because they work and are the best choice.
I have used a lot of the portable pet carriers that my wife and I pick up at various garage sales over the years. I sanitize them and mark the cages with my name, add lots of pine shavings insert 1or 2 birds, depending whether or not it is a bantam or not and then off to the show. I use larger carriers to fit the size of bird, bantam or large fowl. I also have a custom made wooden carrying box, very nice, but has handles on each side vs the pet carriers handles on top. Just a little easier to carry. My thoughts only.
Whoopee, today and tomorrow it will be in the 60's.
Several of the local shows I attend have folks there selling them. They can custom build them to ypur specs or they usually have the boxes sized for the breed. For example: a modern game would need a diff sized box than a cochin. A bantam would need a diff size vs. A large fowl. Consider your vehicle that you drive to the show too. If a 4 stall box is too long u may want 2 2 stall boxes. Or a 3 stall and a 2 stall. I live in NC and have a local guy that shows and builds boxes. He does a good job and they are fairly lite weight.