The premises behind how natural selection leads to the formation of new species are fairly few.

1. Natural processes put pressure of existing species and result in changes to populations.
2. Reproduction of species exceeds the environmental resources that allow maturation to the point of next-generation reproduction (more are born than can live).
3. Members of thus overabundant populations with genetic disadvantages (runts, mutations etc) will be among those that die out before reaching reproductive maturity.
4. Members of thus overabundant populations with genetic advantages will live to reach reproductive maturity and will pass on genetic hereditary traits to the next generation.
5. Significant advantages may be adaptations leading to flourishing (1) in the climate, (2) with the food supply, (3) in intra-species competition (same species), (4) in niche competition, (5) in inter-species competition (other species), (6) in high reproductive rates, (7) in reproductive maturity rate (how quickly they mature), (8) general genetic strength in passing on advantageous adaptations that lead to species flourishing.
6. Population pressure from geological changes results in naturally selected species variation, pressures such as geological pressures that occur when a volcano erupts causing "species to undergo accommodations, migrations, and extinctions" (Charles Lyell, eNotes).

Based upon population success related to these premises of natural selection, populations will show variations over accumulated generations--many generations are required though in the animal kingdom generations are short-lived thus accumulate rapidly--and will lead to visible population variations internally and externally in organs and structure (like finches' beaks). The continuing result will be the intermittent emergence of "incipient species," or new species, such as Darwin found among finch populations in the Galapagos Islands.