I'm thinking it'll continue to let you ride the chocobo as long as you've aquired it with one class.

Many quests in the game let you access them even whilst playing a class of a lower rank.

(E.g/ Rank 30 conjurer, switches to rank 10 blacksmith, but can still access rank 30 side-quests)

So I don't imagine Chocobos will operate any differently.