A bag or a box is the most efficient method of transporting most animals. Wild animals are certainly better in a box on their own, depending on their size and the distance being travelled. They can be put in built-in trailers etc. They are going to do less harm to themselves conscious than they are semiconscious, being sat on all the way down to the vet clinic. Hand-raised animals are often happier in large pouches and just being nursed.
Wombats are certainly better off in bags.
If you have an animal that has to have the vet come out to it, because it is too big or too nasty etc. The animals get used to going up and down along the fence-line between the two fences so they are not worried about it. When you want to catch them up, they come running up along the fence because they've been there before and the person with the pole catches them. A variation on the theme, is to have a race that is there all the time, which runs to a corner, so they can't get out if they run down there. They commonly go up and down there so they're used to it.