Scope, in reference to horses, can mean three very different things. In horse racing, scope is a term used to describe a horse's potential to improve with age. This usage is common in both the UK and international English.
Scope is also used, by way of shortening, to refer to a medical procedure that checks a race horse's airways before a competition. The tool used to examine a horse is called an endoscope, hence the term "scope."
The third definition of scope is found within American jump racing and Australian English. Scope here refers to a horse's ability to leap and clear large obstacles, as in a steeplechase or when jumping for show.