Normally, Handle Animal. If a PC takes "Profession: Teamster" I'd certainly allow it. If there is a Drive or Pilot skill, I'd read it to see if it applies to "all vehicles" or something similar, in which case I default to that. But even if there is a Drive skill, I'd have Handle Animal *also* work.
As a GM, makes sense. My friend is making a sort of M&M-inspired system, but more fantasy-oriented? I think? He's currently working on transportation skill writeups and sorting vehicles between drive and pilot to write his own descriptions. He got hung up on carriages.
Handle Animal was my addition to the question, but I did find Ride interesting. I told him to look online what others had done, but he didn't find much beyond a great and inconclusive forum discussion about chariot combat. I'll add that he wants his game to be able to apply to various settings.