I'm guessing the larger ones had restraint harnesses resembling the slings that aquariums use to transport cetaceans or pinnipeds, or possibly like whatever the various space programs used for animals that they sent up and bothered to bring back. Since the daemons are intelligent the harnesses have a release button within reach. The Wei triplets have it relatively easy; the snake needs a perch to wrap herself around, the monkey gets a setup like one of the space chimps, and the rooster gets something like the pigeon vests used in World War II to secure homing pigeons for transport behind enemy lines.