Thanks all, I'm glad this was useful.
Phil, I wondered about the strength issue, especially on some of the persimmon grafts, which have gone crazy and are longer and more lush than new growth from the main tree. I guess I'd better keep a close eye on them to make sure the union is really thickening up. I noticed the Parafilm and rubber and were already largely deteriorated on one of the fig grafts, so I added a little tape--maybe I should reinforce it some more. In nana7b's recent thread, she showed how she added splints to some of hers. On the other hand, maybe if it's too rigid, it might not add as much new wood. Who knows? It's all new territory right now--the "learn as you go" method.
DesertDance, there are six different varieties grafted onto that one tree right now. Three are thriving, one died (my fault), and the jury's still out on the two varieties from yesterday--but they looked solid and I'm optimistic about them. As far as I know, there's not much to limit how many varieties you can add to one tree--maybe the perfect solution for that disappointing variety that you're ready to yank out and replace. It's a great way to "trial" new varieties without committing a lot of space and irrigation. No risk of receiving contaminated soil, and the growth rate (so far) seems much quicker and more dependable than waiting for cuttings.
Eli, based on the Joe Real tutorial, which is much more detailed and informative than mine, I think you want to wait until it's past dormancy--for persimmons, if I recall correctly, he suggests waiting till the branch has five discernible leaves before cutting it off to make the graft. I jumped the gun a little on the first few figs, but they worked anyway. Just make sure the bark is "slipping" so it's easy to separate.
I can see one reason why it works so well--the rootstock tree is pushing sap with such pressure that within a few days, beads of it were popping up here and there through all those layers of Parafilm--particularly on the cut end, and drying into hard little scabs. Contrast that with rooting a cutting--it doesn't have anywhere near the energy resources, because it has to do all the work. The graft only has to tap into what's flowing up from an existing root system.
One other important point from Phil that I forgot to mention--be vigilant about removing any new buds sprouting from the branch stub. If they're allowed to grow, your graft will quickly slow way down or even die. Keep checking for new, unwanted bud break every few days and rub them out before they can get going.