I heard chocobo race challenges allow you to purchase certain race skills? If that's the case, which one gives Cure III? And what other skills are gained from what races?

I wanted to put off doing the challenges until I got my final chocobo, so I had a source of MGP to fund his feeding. Because of that, I haven't done any of the challenges, and have none of the final tier abilities unlocked. I'm just worried I'd have to wait until the last challenge to get the skill that I want.