I took a quick look through what literature I am familiar with and couldn't find any mention of a practice like this or anything even remotely similar.

However, there are two importance considerations. Horses are tremendously important to the Apache as tools and beasts of burden and even more importantly as markers of status and as a form of wealth. That does suggest they would be wary of squandering such a precious resource by riding it to death.

On the other hand, the Apache are one of the few horse cultures in the US that wasn't at all squeamish about eating horses and made it a regular practice, particularly for importance occasions alongside cattle, but also in daily diet. That said, this willingness to eat horse does need to be tempered by the importance of live horses as both markers of wealth and status and as importance tools for transportation, raiding, and warfare.

I couldn't find any specific references to the activity you are describing in my search of the material, but if you are interested I would highly recommend the book Western Apache Raiding and Warfare edited by Keith Basso.