I did some post-graduate degree research into apple tree grafting. The anatomy of the graft interface was the primary interest. Several practical things emerged.
1. No matter which style (twig, bud, etc.) of graft is done, it is best done in the spring when the sap flows and the bark slips easily on the cambium secondary meristem.
2. The wound has to be covered to prevent dehydration and the entry of potential pathogens such as bacteria and fungal spores.
3. Gentle pressure needs to be applied to the fresh joint to prevent the growth of a mass of non-woody cells to weaken the connection. This accounted for most mechanical failures.
4. Do your best to match the cambium growth layers.
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So how to do these things? We used "grafting rubbers." They were simple strips of latex rubber, about 7mm wide by 12 cm in length. Several wrappings and a finish similar to a half hitch. Inherently, just about every thing gets done with a single effort. Being latex, the material oxidizes from oxygen and UV, goes brittle and outdoors, falls off in a couple of years.
I've seen much more modern plastic strips but I'm not impressed with the lack of elasticity. I suppose that horticulture supply houses are the first places to look.