Well when a species transitions it still can breed with it's "parent" species (not in all cases but most cases it happens this way) so then you get a hybrid between the new species and the parent species. That offspring may in turn breed with it's parent species or breed within it's own.
So for humans for example.
When the new species of homo sapien was derived there was still breeding happening between homo sapiens and homo heidelbergensis which created a hybrid between the two.
So you in fact never get a pure species UNTIL that species becomes genetically locked, either by a change in location, food source or a new disease destroys the parent species or biology where the species can no longer breed. In other words the transition from one species into another is a very gradual process with the old mixing with the new until something prevents it.
As you continue back through the branches of evolution you get this constant crossing of old mixing with new which diversifies the gene pool. So you never actually come to a point where a species only has two parents that give rise to an entire species.
We have seen this in researching cheetahs. There was a point in it's evolution where it's gene pool became very thin and it almost was wiped out because of this. When comparing the genetic diversity with other large cats it's gene pool was very small. They estimate the numbers were probably around ten thousand and had it dropped any smaller it would have reached a no return point because it would have resulted in genetic defects.
This narrowing of the gene pool makes them extremely close in relation to one another. Where if you do DNA testing over large areas the results show they are related even though these packs never actually interact because of the distances between them.
When forced to breed in captivity many times the offspring have genetic defects because the parents are so closely related the resultant genes give rise to massive defects and abnormalities. They are on the endangered species list not because of their beautiful coveted skins but because their offspring often have defects that prevent the offspring from surviving.