Saying, "Darwin admitted this or that," is a poor argument, since he was around in the very infancy of the idea of evolution and has nowhere near the evidence we do now.
And the long period of time was referring to the rarity of a mutation, not the transition species themselves. The temperature on the Earth has varied so much over the many years that many species not suited would end up extinct. Things like ice ages and mini ice ages, as well as global warming, and when I say global warming I'm not talking about now. There have been many times of global warming in the history of the Earth. These are just some of the examples of how much the environments of the world have changed that could potentially cause the extinction to some species but not others.