As has been pointed out, it isn't so much a question of how much gets rearranged (because everything is potentially shuffled) but is really a question of how much is lost or changed, since any genes/alleles present are going to be subject to expression. This is a simple Mendelian genetics problem, but to really answer it you need to know whether your alleles are in a heterozygous or homozygous state. If you take locus A, for example and cross Aa x Aa, you will find the next 25% of the progeny will be AA, 50% will be Aa and 25% will be aa. In essence, you lose 50% of your heterozygosity with each generation of selfing. Why is this important? If you are highly heterozygous to begin with, you will have difficulty recovering selfed progeny that resemble the parent. The more homozygous the parent is, the more likely you are to recover selfed progeny that closely resemble it. For crops like peaches, which are self-compatible and descended from a narrow genetic base, that means that lots of important alleles are more likely to be homozygous to begin with and less will be lost from each generation of selfing, whereas some other type of crop which might have higher levels of heterozygosity (an F1 hybrid strain or species that evolved as an outcrosser and/or has a wider genetic base) will be quite different.
For something like Prunus tomentosa my guess is that "domesticated" clones in circulation are probably all descended from one or a few individuals. If they tolerate selfing it is quite possible that seedling strains have come about over the last Century (either intentionally or just because people found it easier to grow out seedlings than to propagate clonally on a large scale) since it was introduced. Since each generation of selfing leads to loss of half of heterozygosity, you can very quickly reach a situation where alleles for major characteristics have become fixed in a small population. Thinking about it this way if you start with a scenario where EVERY gene is heterozygous, in one generation only half will be, in two generations only 25% will be and with three generations of self pollination only 12.5% of your genes are in a heterozygous state. At this point you can see that just 3-5 generations of selfing can lead to fixation of most of the alleles and that if only one plant is saved and moved forward from each selfing event as "the best" you will have something that will give progeny that are phenotypically very similar to the selfed parent. This doesn't always work from the standpoint that many plants don't tolerate selfing and suffer from inbreeding depression, but this is usually related to the breeding systems that they have evolved with and species that evolved as self-compatible in the wild tend to be more tolerant than those which largely evolved as self-incompatible.