Graft Compatibility Between Different Species
You are going to run into the least amount of problems when you graft scionwood onto a member of the same species, but it is not uncommon for those that are closely related (i.e. members of the same genus) to have some level of compatibility. For example, peaches are generally grafted to a peach rootstock, but to reduce vigor or ensure that the roots are hardy enough for marginal areas, certain species of plum can be used as a rootstock for at least a few varieties.
Alternatively, rootstock from another species may frequently be used because it offers more benefits for a particular region. This is generally the case when it comes to apricots in North America. Compatible varieties are grafted to the rootstock of a peach or a plum, if not some kind of interspecific hybrid, but in colder areas, seedlings from a Manchurian apricot (Prunus mandshurica) may be used. Conversely, true cherries are unlike the others in that only in very rare cases can they be grafted to the rootstock of another stone fruit from this genus (and vise-versa).
The Myrobalan plum variety 'Adara', from the species Prunus cerasifera, is the most notable exception, and – in addition to cherry – it is compatible with peach, plum, apricot, and almond, although it will depend on the variety (particularly for apricots and almonds). Adara is sometimes propagated as a rootstock for use in heavy clay or calcareous soils, but it is difficult to find in North America. Rather, it is occasionally distributed here for use as an interstem, or an interstock as it is sometimes called, which is scionwood that is used to connect a variety and a rootstock that would otherwise not be compatible (by grafting it between the two). The interstem would, of course, have to be compatible with them both.
Interstems are also used to inhibit the growth rate of a vigorous rootstock and increase precocity. For example, a standard rootstock with an interstem from a dwarfing rootstock will often create a semi-dwarf tree, but the length of the interstem may determine how great of an effect it will have. While I have not seen a source that demonstrated the most feasible length, I recall six to twelve inches being desirable. Additional length as well as grafting closer to the soil or even partially burying the interstem (the latter of which may prevent suckering), can amplify the dwarfing effect, but this effect would also reduce root mass, thus causing some trees to benefit from staking. When an interstem is used for the purpose of resolving compatibility issues between the scionwood and the rootstock, it seems that roughly four inches or so is enough.
In some cases, you can graft a member of one genus onto a member of another, provided they belong to the same family. One example that comes to mind is the relatively new combination of European pear scionwood (Pyrus communis) and a juneberry rootstock (Amelanchier spp). This combination is quite notable because European pears generally take a very long time to produce, and there are only a few pear rootstocks that induce precocity. Those that do are only semi-dwarfing at best, but juneberries can be used to produce much smaller pear trees that are very precocious. In addition to these advantages, the juneberry has a few others over the alternatives, such as cold tolerance and resistance to fireblight, but improved selections are, for the most part, not yet available to the public and, like the quince, may have incompatibility issues with a number of varieties. This can, of course, be remedied with the use of an interstem, such as Comice, since it appears to be compatible with multiple juneberry selections.
In rare situations, an unusual amount of difficulty may be experienced when grafting members of the same species together, such as with chestnuts, but this problem is not necessarily the result of genetic incompatibility. A special technique is required when grafting any combination of chestnut to avoid a high rate of failure, but genetic incompatibility does, in fact, become quite significant when mixed selections are used (in-spite of the ease of which the various chestnut species hybridize with one another). Graft failure may not be immediate as well. It often takes a few years to occur, in this case. The situation is dire enough that nurseries that sell grafted chestnut trees are becoming fairly uncommon. Instead, they sell seedlings from the species 'Castanea mollissima' (the Chinese chestnut), since those from improved selections often produce nuts of an acceptable quality.
Of those I have grafted, my favorite combination is currently the European plum (Prunus domestica) with a Nanking cherry rootstock (Prunus tomentosa), and even though the Nanking cherry has a flavor that is reminiscent of a tart cherry, the reason why this works is that it is genetically closer to that of a plum than an actual cherry. Altogether, there are four European plum varieties and one Asian plum grafted to it, and they appear to be doing fine.
While I am sure that there are better alternatives for my climate, I find this combination to be quite intriguing because European plums are known for taking at least a year or two longer to produce than most other species and there are only a few rootstocks for it that are known to induce precocious behavior. The Nanking cherry is supposedly one of them, and that seems to be the case for me. In fact, the bush is in a very poor location that receives a significant amount of shade, and it still caused some of my grafts to produce a good amount of plums on their third leaf, which is at least one year earlier than those I grafted to a well-established European plum tree with good sun exposure.
Related: Introduction to Grafting
1. Physiological, biochemical, and molecular aspects of grafting in fruit trees
2. Adara, A Plum Rootstock for Cherries and Other Stone Fruit Species
3. Effects of different interstock length on vegetative growth and flowering in peach cv. Pavie Catherine
4. Apple Rootstocks: Interstem Apple Trees Revisited
5. A review of recent Pyrus, Cydonia and Amelanchier rootstock selections for high-density pear plantings
6. Graft Compatibility among Chestnut (Castanea) Species
2. Adara, A Plum Rootstock for Cherries and Other Stone Fruit Species
3. Effects of different interstock length on vegetative growth and flowering in peach cv. Pavie Catherine
4. Apple Rootstocks: Interstem Apple Trees Revisited
5. A review of recent Pyrus, Cydonia and Amelanchier rootstock selections for high-density pear plantings
6. Graft Compatibility among Chestnut (Castanea) Species