When reading historical fiction and non fiction, it’s a relative common occurrence for riders or carriages to change their tired horses for fresh horses at post stations (or inns and other establishments) to enable a higher speed to be maintained throughout the journey. For stage coaches that make a specific journey repeatedly, one could assume they have a string of horses spread down the route and switch their own horses out for fresh horses that they own. Likewise for military supply and messenger routes. But for private individuals, who owned the horses they switched out their own for? in switching out the horses did they forfeit the horses they arrived on, even if they were better horses? Or did you personally have to possess horses at a post station in order to make a switch?
The horses were owned by the company that ran the line - the Pony Express had a series of stations that would provide a fresh horse for rapid change. The exhausted horse was then tended and prepared for the next ride. By shifting them east and west, stations maintained an adequate inventory. Similarly, Stage lines that traversed long distances would have stations where company horses could be switched out for exhausted animals.
Private riders would not normally surrender their animals. They usually rested at a station until they and their animal was ready to move on. Sometimes - although it was rare - stations would trade out animals, but that would be an expensive proposition for the rider.
Mormon settlers at the eastern foothills of the Sierra Nevada range famously sold fresh oxen to people crossing the continent in wagon trains. Their beasts were often too exhausted to make the climb up the steep Sierra escarpment. Having typically arrived at the end of autumn, the emigrants needed to hustle across the mountain range to avoid becoming the next Donner Party, so they would buy fresh, well-fed oxen, trading their own for next to nothing. The Mormons then tended the worn-out oxen during the following year and sell them to the next batch of emigrants. This sort of arrangement was an anomaly, however.
I think you're asking about traveling post. I'm no expert, but my understanding is that you would hire new horses at each stage of your journey, with the postilions either getting hired by someone else for the return journey to their station, or just taking the horses back themselves. The horses would be owned by the person who ran the posting station, who also employed the postilions. The post chaise could be hired as well, in which case you wouldn't have horses of your own to worry about. If it was a private chaise, my understanding is that you would either board your horses at the first posting station until your return, or hire someone to return them to your home after they had rested. I'm having no luck finding a source for that, so I'm not sure if my opinion is based on something I read at some point but forgot the source, or whether it's my own assumption. Hopefully someone else can add more information. Again, not an expert, but some quick googling yielded this: https://www.thetrafalgarway.org/post-chaise-travel https://www.britannica.com/technology/post-chaise https://candicehern.com/regency-world/glossary/post-chaise/