They can be trained and will obey commands, such as "Away" (making them leave the area) even during the heat of battle. They can bear the weight of a Shardbearer easily and can communicate with them. They barely require 'handlers'.
According to Gavilar, it is rare that they roam far east. Further, Dalinar sees Musicspren trailing after them in the air.
Besides being larger than regular horses, Ryshadium also have a few other unique features. They have stone hooves that are stronger and more durable than a regular horse's hooves, and never need to be trimmed. They also have a large rectangular pupil in the center of each their eyes.