Well in some films, like The Lord of the Rings, the principal actors that had a lot of on-screen riding (ie. Karl Urban as Éomer, Bernard Hill as King Theoden, Orlando Bloom as Legolas, etc.) had riding lessons, and they were able to do the majority of their riding work.
However, in many films riding doubles are used. Going to the Lord of the Rings examples, as many people have seen those films, Liv Tyler had Jane Abbott as a riding double for many scenes (ie. the chase to the Ford), as did Ian McKellan as Gandalf, mainly because of the nature of the riding they had to do, which involved quite a bit galloping and stuntwork.
From Horse & Pony: Lights, Camera, Action … Joan Gilchrist visited the equine stars of “The Lord of the Rings.”
Jane Abbott & Florian (Asfaloth)
Jumping through hoops
Jane & Florian
Lord Of The Rings Equine Star Stars Again
The Florian and Jane Adventure
If you add the cartoon back to Brego page, here’s the cartoonist link: http://sevspace.com/stupidarchive/fotr.asp.htm