Leases.
Assuming you have permission to lease a horse (e.g. you own it, it’s not currently for sale/lease), you will see an Offer for Lease button on the right sidebar of the horse page. That will give you options to lease the horse.
When leasing a horse, you can request to cancel the lease, and undo the cancellation if you change your mind. You can request a refund, which will be processed if both parties agree to the refund.
Alerts haven’t been added yet (for new leases offered to you, or current leases being cancelled), but will happen eventually.