I've seen the discussions before, so I won't rehash them here, except to say in brief that it is matter-converted-to-energy vs. "wormhole" notion of beaming. (In the former, one would expect some sort of imposed "stasis" as the conversion takes place. Movement would result in the disgusting "slicing" effects described in Michael Crichton's Timeline where micro discontinuities between nerves and tiny blood vessels occurs. It wouldn't take too many passes to completely mess someone up, vindicating McCoy.)
Further up-thread I compared the "jumping" in Steven Gould's book Jumper with the visuals in the movie. That was partly to make a point about the different looks of the phenomenon. But the book was clear about what was happening. Davy, the jumper, is experimenting with the aid of his girlfriend, Millie. The first test is simply a video recording of a jump. This shows a single frame after Davy disappears with a "Davy-shaped-hole" with the background of the new location. He's creating a hole and pushing himself through. (Although he does not feel it as motion.)
In other tests, Davy handcuffs himself to a wrought iron rail, then tries to jump. This painfully sprains a shoulder, and Davy is kept from completing the jump. Then, free of the cuffs, Millie tries to hang on to Davy. He slides away like a greased melon seed. They try again, this time Millie using a wrap-around wrestling hold, and she goes through.
In the second book, it is discovered that the extremely brief "hole" can allow water, or higher air pressure to slip through. He uses this knowledge to develop a new technique that he calls "twinning." He jumps back and forth, only a meter apart, faster and faster until there are "two" of him facing each other. They even reach out and feel the ghostly touch of hands. Twinning becomes a powerful tool. Davy discovers other fascinating techniques in other books.
