As multiple people already stated in this thread, after you have passed on enough stats through generations, your little graph on the animal selection screen will have a red outline of the white stats. The new combined stats are what you play with the next time you load that animal.
Edit: Also I have personally tracked stats and verified this in-game on multiple animals, in case you still doubt. Example below:
G1 Animal
HP - 50
Hung - 50
Sta - 50
G4 Animal (stats upon breeding not when dying)
HP - 55
Hung - 53
Sta - 60
If this was say your first playthrough with some animal, the next time you load up a New Game with that animal, you will start with the G4 stats listed above, not the G1. Also, this is why it's extra beneficial to find Prime mates whenever possible as they will pass on the most amount of stats earned from challenges.