I had a free lease on a mare that I eventually bought. She had been donated to a therapeutic riding program, but was more suited to able-bodied lessons than to therapeutic lessons. She was used only a fraction as often as they had expected, hence was more of an expense to the program than they could afford to carry over the long term. I had the free lease for 8 months or so, maybe a year, because she needed to be exercised and I was happy for the horse time. When she was ridden consistently, they were able to use her in more lessons, so it was a win-win for everyone. She didn't cost me anything at the time, she was truly a free lease. That meant all her expenses were still paid by the lesson program, and all vet/farrier/feed/tack decisions were made by them and I didn't have input into that sort of thing. Sometimes I would help out by holding her for the vet or farrier, but even that wasn't really necessary because she was on the same schedule as all the other horses owned by the lesson program. The only things I spent money on were "extras," like a new halter or a blanket. When they eventually decided she wasn't going to remain in the program because she still wasn't being used enough in lessons, they offered me first chance to buy her. I did, and they agreed to let me continue to board her there at the same rate as other outside boarders.
I don't know if it's as easy to find a free lease now as it was then (12 years ago now). I do often see posts about horses needing more ride time on our local horsey Facebook pages. Generally those are seeking a rider who will work with the horse at its home stable because the owner has too many horses and not enough time. My guess is your best chance at finding a situation like that is to network via existing lesson programs, or by volunteering at either therapeutic riding barns and/or at horse show venues. Basically, find the places in your area where horse people congregate, and just put the word out that you're looking. If you are taking lessons, you'd probably have the bonus of people who might be willing to lease a horse seeing for themselves whether or not you can ride the way they'd like their horse ridden, and trainers/instructors often hear from people with a horse that could be available to the right person.
I don't have any experience with paid partial leases, or off site leases, but I know there are other people on here who have done that and can share their experiences.