In the past i used cubes in seed trays in a duffel bag with the lid taped on, could carry 24 seedlings in at once.
If you are using cups then just tape a cup ontop and you are good. Its easier if they are straight sided and square for carrying as they wont move so much but you can always just pack a shirt or something around the bottoms of them to stop movement. Backpack On your back creates a lot more bouncing but it isnt that bad to be honest.
You can always cut tops off 2L drinks bottles and slip that over the cups too. Theres a million options.