A single baseplate with tiles weighs about 1 pound. It's really hard to find a cardboard box that can survive very long, even filled with only 30 of the boards.
Last weekend, my father-in-law helped me build some very nice wooden boxes that I can use to transport the baseplates from show to show.
They are designed to stack, to make it easy to transport and store them.
Today, I'm taking them to their first show. We'll see how well they work.