So I've been testing my 622c with my 5D Mk III, and two lenses - Tamron 24-70mm f/2.8 VC, and my nifty fifty, dark room with a full press firing a flash (but no pre - the only light in the room is the 622c AF assist light), but trying a combination of full press, and half press focus lock confirmation then full press.
Both claim to lock focus in a second or less when using the 622c (i.e. they beep). Only the nifty is actually nailing focus consistently though. And the Tamron ONLY struggles when I am using the AF assist light from the 622c on top of the camera, and only when the subject or point of focus is several feet or more away. It tends to backfocus a lot (by several inches), though it apparently got bored and decided to shift to front focus for a bit of variety a couple of times. Within a few feet of subject, it nails focus every time on just the 622c AF assist light.
Ironically, my 60D with the same lens and 622c nails the focus every single time. It's quick, it's accurate - very close to the same as shooting with either body with continuous lights on (with continuous lights on, the back/front focus issue on the Tamron simply doesn't appear).
I don't know how the 622c/5d Mk iii lens is going to react to different lenses is the bottom line. I know it doesn't play well with my Tammy when I absolutely have to have the AF assist to focus at distances of several feet or more (makes me wonder how the 70 - 200 II would be with this). Otherwise, I can shoot several feet in front of me with no problems. Or I can put it on my 60D on the few occasions where I'm in that dark of an environment and I need to be able to back up without going wide.
Good thing is a little 622c doesn't cost that much. I wonder how the Canon 24 - 70 II 2.8 would do with it on a 5D Mk III body, if anyone is able to test I'd be very curious to hear what your experience was.