Then how can a trans-muted species EVER come into being? If cross fertilization does not produce a fertile offspring to propagate itself then where is mechanism for continued evolution?
I don't get the feeling that you're very familiar with models of speciation. It's about isolated subpopulations going their own way.
Cross-fertility is simply the most common test for whether a new species has arisen. When you've lost it, you have a new species. Hybridization can produce instant speciation in plants, but that's not the mechanism by which, say, humans diverged from apes or amphibians diverged from fish.
Here's a scenario:
Speciation by Punctuated Equilibrium.The Short Summary
A group of creatures gets isolated from the rest of their species. They can evolve easily, because they are a small group. Later, they spread and replace their parent species. Examples are known.
I can be a little slow. Finally, finally I understand your answer. The mules aren't the new species. As you point out, they aren't fertile with anything. The only reason there are mules around is that people keep interfering with the natural mating preferences of the animals involved.
That horses and donkeys can produce mules shows that they are far more related than, say, horses and caribou. The most common case is that different species, if they can be induced to try to mate (or artificially inseminated) are totally cross-infertile. That the mules are themselves are infertile shows that horses and donkeys are no longer the same species.