The possession, display, and gifting of “exotic” animals was a constant of the medieval and early modern elite from England to China. Unfortunately, sources don’t provide the clearest picture of the mechanics of this social trade. As you can imagine, chroniclers were much more interested in enumerating the splendor and exoticness of the massive (and minor–Matthew of Paris loved him some porcupine) beasts. Still, every now and then we catch some glimpses, although more pertaining to the upkeep of animals than their original capture.
Texts of veterinary science from the Arab world outsource the problem. While these books offer solutions for what ails common animals (especially horses), when it comes to elephants and “special” donkeys, a.k.a. zebras, they mention the importance of having a trained keeper arrive with the animal from India or sub-Saharan Africa. Europeans used outsourcing, too. The lions who arrived in 1775 London from Senegal were proudly presented by English soldiers–who had stolen them from the original trappers and then killed the people.
Medieval people were well aware of the art of luring, and also of provoking. Jean de Joinville tells a tale of a lion hunt in Caesarea (popular among crusaders), which might offer some insight into trapping:
The King set to work with his people to hunt lions, so that they captured many. But in doing so they incurred great bodily danger. The mode of taking them was this: They pursued them on the swiftest horses. When they came near one they shot a bolt or arrow at him, and the animal, feeling himself wounded, ran at the first person he could see, who immediately turned his horse’s head and fled as fast as he could. During his flight he dropped a piece of his clothing, which the lion caught up and tore, thinking it was the person who had injured him. And while the lion was engaged thus the hunters again approached the infuriated animal and shot more bolts and arrows at him. Soon the lion left the cloth and madly rushed at some other hunter, who adopted the same strategy as before. This was repeated until the animal succumbed, becoming exhausted by the wounds he had received.
In most cases the point was surely the lion’s death, but one can imagine a similar bait-and-subdue method used for capturing. Medieval bestiaries (texts describing the traits of animals, moralized for religious instruction) are not known for their rigorous devotion to realism, but this English bestiary illumination shows goats instead of humans used to lure the lions, who are then trapped into a pre-prepared hole or ditch.
Medieval animal-keepers were well aware of the use of chains and muzzles to control their charges.
Gaston Febus in his Livre de chasse divides animals into two groups: those who dogs “commonly and willingly hunt”, and the rest of them. That group, notably, does not include the great exotic beasts. Nevertheless, the English monarchs, at least, took great sport in using dogs to lion-bait for their entertainment.
Transport would have been situationally dependent, as the journey of the 18-19th century giraffes show. The giraffes headed from Sudan to Vienna and Paris seem to have been transported the same way from the African interior to the coast, strapped to the backs of camels. At that point, they were transported by ship to Europe. The giraffe bound for Vienna was carried across the Alps in a cart, a journey which ultimately damaged its skeleton and internal organs enough that it died soon after arrival. Zarafa the Giraffe, headed for Paris, did better. Her keepers elected to land in Marseilles and have her walk to Paris, protected from weather by an oilskin blanket.
As far as transport and keeping goes, the biggest logistical issue was feeding the animals! Most of our menagerie records, in fact, concern either the money used for food or the amount of food. The (probable) polar bear in 13th century London, when kept on a “one long and strong cord,” was allowed to fish in the Thames for its meals! It seems that many of the carnivorous beasts in the London menagerie ate primarily sheep. Much later, Zarafa the Giraffe traveled with 25 cows to provide her daily requirements for milk. (Apparently giraffes drink milk?)
We should not understate either the skill of medieval animal-keepers–or the inherent danger. Arab rulers imported keepers with their beasts for a reason; English kings designated “Master of Lyons and Bears” for a reason (even if they did not always pay on schedule). They reached certain levels of taming with their charges, who nevertheless remained wild animals. A tragic example from 17th century London demonstrates this:
Mary Jenkinson, living with the Person who keeps the Lyons in the Tower, going into the Den to show them to some Aquaintance of hers, one of the Lyons (being the Greatest there) putting out his paw, she was so venturous as to stroak him as she used to do, but suddenly catched her by the middle of the Arm with his Claws and mouth, and most miserably tore her Flesh from the Bone…They thrust several lighted Torches at him, but at last they got her away…She died not many Hours after.