The problem you describe is obviously a threat to Aristotle's view of nature and of the species of plants and animals (which may be why Aristotle argues against Darwin in Book 2 of the Physics). As you say, "species are organic processes" -- although you ought to recognize that this conception of biological species is our shared conception of species after Darwin. Darwin has indeed made many elements of the ancient theory of nature hard to imagine, even if the ancients found their view of nature extremely easy to imagine.
Plato differs from Aristotle, however. For one thing, Plato expects to find much less order in the natural world than Aristotle does. If you confront Plato with the spectacle of constant change in nature, he might be inclined to agree. In this particular case, a lot depends on whether or not Plato thought there were Forms for species -- a Form of human being, of dog, of oak. In some of the dialogues that speak of Forms, the description of them does not seem to include biological species; however, an open-ended discussion in the Parmenides suggests otherwise, that Forms would have to include species. The textual evidence is inconclusive. And if there are no Forms for dog or willow, the changes in dogs and willows that we see are no threat to Platonic metaphysics.
It's possible to go a bit further than this, though not to the evolution that we know after Darwin. Plato's Timaeus proposes one kind of evolution that ancient author sometimes found acceptable, namely a "devolution" to a worse kind. This is a version of the idea that humans couldn't come from apes, but that thoughtless, brutal human beings could degenerate into apes. (Mind you, the distinction makes no sense in modern biology, because the theory of evolution does not consider us higher than gorillas, or even higher than plankton. I am trying to speak as the ancients would have spoken.) And in the Timaeus, we find something like a cross between degenerative evolution and reincarnation. Male human beings who do not comport themselves virtuously enough are turned into women, and humans are turned into animals. You can't make yourself a better creature on your own, but through vicious thought and action you might be made into a worse species.
For a dozen or so reasons we are likely to find this proposal unacceptable -- morally, scientifically, metaphysically. But it does sound something like evolution. And it is Platonic. So maybe it's the best evidence for the conclusion (in response to your opening question) that evolution as such is not a problem for Platonists.