I wanna say get a set of foam floor mats for the board: the sort that come in interlocking squares that are 2cm or so thick by 20cm or so on a side (I don't really know the correct measurements, but you know what I'm on about, right?). You could then break the board down and stack the squares in waterproof plastic storage tubs with a desiccant jar.
I've always like fake fur mats, even though I don't have one. I feel like with a cloth or fake fur mat and a thick foam mat underneath, you could probably put trees and other similar terrain bits (phone poles, mail boxes, scaffolds, whatever) on wire spikes instead of bases for a more seamless look. Maybe glue grass tufts and shrubs and the like around sewing pins.
Maybe check out artists/architects storage/transport tubes for the mat. Run a bead of silicon calk around the cap to make a gasket (if it doesn't come with one), and toss some desiccant in the bottom. Could probably DIY a sealed tube with a screw-off end using PVC or ABS pipe, but it would be heavier to lug about.