Oh, good thread. I just ran across it.
I have a parrot. I haul her around sometimes in my trailer. She's got a small travel cage (small enough to fit under an airline seat). She goes in the cage, and the cage goes into the Burley Nomad. I use some towels to keep the cage in place in the trailer.
I usually put her cage in the rear of the trailer, and then open up the nylon top of the trailer back there. The small travel cage has an open top, that I partially cover with a towel. My bird can then look up and out to the sky or the surrounding trees if she wants to, or she can retreat under the towel.
All in all, she likes riding around. She gets some views of the outdoors, fresh air, sunshine, and some chat with wild birds. At the same time, she isn't just out in the open. (As with nearly all pet birds, being under an open sky makes her paranoid.) Often, we go to fun places: a friend's house, a nearby bird breeder's, a park, the woods. So, she's very willing to enter the travel cage, and gets excited when she sees me getting the bike equipment together.
When she was younger, I used to just sit her on the handlebars and go riding. If she fell off, she could fly just enough to catch up, and grab onto my shirt. Now, she's a better flyer. If you know about parrots, you know that even though a parrot would prefer to stay with you, she can be prone to panic-fly. If that happens, you might not ever see your bird again. So, no more unrestrained activity outdoors for my bird now.
Anyway, some lessons: with birds, and undoubtedly with other animals, getting them used to some sort of restrained travel at a young age is a *great* help. If travel is fun-- that is, not always to the vet-- it's much easier to elicit the animal's cooperation. Partially covering the animal's enclosure can help provide a feeling of security. And, as the previous remarks assume, I think it's best, if possible, to enclose the animal in a small travel cage before loading her into a bike trailer. This is more secure for the animal, safer if there's a crash, and can ingrain habits that would be just as useful in getting the animal comfortable with riding in a car, too.
A cat can fit in a small travel cage. I've seen people take cats on airplane cabins. If at all possible, I'd try to accustom a cat to entering a travel cage before entering a bike trailer.
Good luck with your non-human friends.