Wow- spent my morning following all of the posts on this thread.
Found some old songs and some new ones here. It is unfortunate that some of the leads went nowhere, because I was touched by a number of songs I found.

I remember Wildfire from my youth and how much I felt that song at the time.

Home, Lads, Home- has been on my list to learn for some time now. Like others on that list it moves me so much I'm not sure I will ever be able to get through it without breaking down.

Horse Days- tune? could not find this anywhere.

The old Ways by Mike truelove- I can't find anything about this song, and would like to learn it.

Found Archie's rendition of the Last Clydesdales. Will add it to my list, along with Blossom the Mining Horse.

I probably have a few to add myself, though I am mostly focused on songs that are really all bout the horse(s). So The Tinkerman's Daughter (traded for a pony) and similar aren't on this list.

Hopefully others will be inspired to add to this thread...