I don't have anything to add to the ideas offered by other cat-wranglers. (We use the hind-end first, gravity assist method)
My coworker went to a...cat acrobatics? tricks? type of show where this lady had trained a herd of cats to do all sorts of tricks, and they more or less were obedient to her commands for running an obstacle course or leaping/flipping etc, but would occasionally wander off into the audience. She wasn't worried about that though, because apparently, one thing that they were SOLID on was going to their carriers when the signal was given. She said she recommended at least that bit of training for all cats, because there are a variety of emergency situations where it could be useful as well as the ordinary vet visits. Apparently, with the right treats, cats can be clicker trained similarly to dogs. (I am still skeptical)