Yes. It's called natural selection, and it's one of the main driving forces of evolution, the other being genetic variation, which provides the raw material upon which evolution acts. In the case of your plants, a small "founder population" (maybe one plant) with a limited genetic set becomes established and, absent genetic interchange with plants in other populations, it can establish its own variety or even, eventually, its own species. However, pollination and seed dispersal (insects, wind, bird droppings) can be effective over many miles, preventing true genetic isolation for the new population unless there's a very effective barrier between them.