I'm a novice myself just trying budding for the first time this season, but from my understanding, budding should absolutely work with both mulberry and persimmon.

It only takes a short while (say two or three weeks), for the tree to completely callus up around the bud (probably because the wounding is relatively minimal with this method), so budding can be done quite late into the growing season. The new bud does not begin to grow until the spring anyway.

According to the literature, the growth above where the new bud is inserted should be left on until the bud itself begins to grow, in other words, the spring following grafting. But you should know if the bud had taken just by its appearance in the weeks after budding. A plump green bud that quickly and cleanly drops the old petiole below it is likely to be viable.

Hopefully the really experienced folks will chime in also...

You can probably still do 'em through the rest of this month, but at this point, I'd probably wait 'til next spring to force 'em.

Thanks fab and lucky for the feedback. It was just the nudge I needed to try some bud grafting. I never bud grafted anything until the last week. I started with mulberries and persimmons. Time will tell if I did anything right, but I'm happy to have started practicing and learning. I really like the idea of late summer budding now that I've tried it.