For larger birds I use small animal cages, like rabbit hutches. Depending on size of birds I will make judgment call on how many it can carry, trying to make sure they can all lay down/squat. Problematic part is feces. I have a designated 6x6 tarp for lining under the cage(s). When done I shake it off, lay outside and let the rain wash it off. For smaller birds, small enough that they can escape the cages I just use a box of appropriate size to accommodate chicks. Lid is loosely shut or folded together. Haven't ever lost a bird. But I only travelled an hour at max.
I knew a farmer that loaded 100 white leghorns (From an Amish laying house being replaced) into a Ford Taurus in feed sacks, 5 birds per sack. Trunk and back seat filled and lost 5 birds out off 100. I thought for sure he was gonna lose at least half. Not suggesting this but chickens are rather hardy and can withstand a lot of abuse.