Regional business model has kind of split between two types of operations...

Some regionals just operate planes owned by mainline. Priority there is a low cost of operations, contracts are short, and planes get moved around to control costs.

The traditional model, where the regional owns the planes, allows the major to purchase capacity without having to commit to owning/leasing planes. Costs a little more, and the regional won't buy new planes without a lengthy contract, normally over ten years. Once the initial contract is done, the planes might get extended, contracted out to another major, sold, etc. Major can't fire the regional unless they're willing to give up the planes too.

Some regionals do both models. Mesa is a good fit for the former model... low cost, but you can take your planes back in a hurry if/when they fail to perform.