An equine disease would surely affect all horses? The impact would be upon the artillery and the transport train rather than the cavalry. Maybe cavalry would have less horses suited to their wild charges but they can carry out their other (and arguably more important) duties on smaller horses but guns need strong horses. Oxen would be a more robust alternative to horses for artillery but, whilst more certain, are slow. One might look to India though where the bullock cary determined the speed of a major all arms force. This left their cavalry free to roam about and return to the slow main train.
In the South African War it seems to have been a cynical decision that it was logistically easier to replace horses than to keep mobile mounted infantry supplied with enough fodder. It was one of the claimed virtues of bicycle mounted infantry that they were equally as fast as horse mounted infantry but needed no fodder and troops were spared caring for horses so were better rested despite the pedalling.