Bullfighting on horseback is a traditional Portuguese style of bullfighting practiced in Portugal, Spain, Mexico, and California. In the Spanish world, the term used for a bullfighter on horseback is Rejoneador (rejoneadora for female) rather than Cavaleiro (cavaleira for female).
Bloodless bullfighting is practiced only in California. In Portugal, the bulls are not killed at the end of the bull fight, but blood is drawn during the performance to slow and tire the bull, that is why it is not termed as a "bloodless" bullfight in Portugal.
In California, there is no blood drawn and the bull is definitely not killed at the end of the fight, but they are sent back into the pasture to either breed or graze in the fields.