As this was the job I always wanted, I know plenty about it. I live in Dallas and have spent some time talking with their mounted unit. In Dallas, you would go to the academy, spend 2 years (or more) on regular patrol, then you can apply for mounted, it is a highly sought after position, so not too many people make it. However, DPD is paid mounted, not volunteer, and I believe they are all full-time mounted, no longer work car patrol.
I lost hearing in one ear when I was 20, and therefore am not eligible as a police officer, so I can't do mounted, but I sponged up that info.