Chocobo Caravans are divided by Grand Company, and you are specifically told you can not change Grand Companies. We were told many times when Grand Companies were implemented that once you pick one, you are stuck with it. Rewards are comparable across each GC, so join the one you want.

Also, in the FFXIV Play Guide under the Grand Company Guide:

Caravan Escort Duty

Caravans hauling materials borne on the backs of chocobos support the bustling economies of the city-states. Safely escorting these caravans is one of the Grand Companies’ most important duties.
NPCs near aetheryte at the locations listed below periodically recruit adventurers for caravan escort duty. Speak to an NPC recruiter to participate and then head over to the caravan. A maximum of four adventurers can participate in a single escort.
Once the caravan leaves, enemies will target the cargo. You will need to protect the caravan from ravenous enemies while issuing appropriate commands to the chocobos in order to arrive at the destination before time runs out.

* Recruits cannot undertake caravan escort duty.
* Vanquishing large numbers of enemies while on duty may result in rewards and bonuses upon duty completion.
The Maelstrom Camp Bloodshore / Eastern La Noscea (36, 20) Friont
The Order of the Twin Adders Camp Tranquil / South Shroud (38, 49) Marquaile
The Immortal Flames Camp Horizon / Western Thanalan (13, 29) Mourelz
* Adventurers cannot accept caravan escort duty from other Grand Companies.