Precognitive and premonitory dreams come from the unconscious, that exist in a different realm, which may be beyond our time and space. Usually only the bad events like illnesses, accidents and death can permeate our membrane to cross over to show themselves. One of the factors that enables the crossover can be thin times, which are special days or moments in time that are closely connected to the person concerned. When the future crosses over to inform the dreamer, the dream becomes precognitive and premonitory.
The Universe is natural and its symbols are usually natural elements. To represent a mode of transport, the symbol expressed and shown is usually a natural mode of transport like a horse, donkey, or camel where a human can ride upon. These animals for transporting humans and are of transport themselves, are associated with carrying a rider far away beyond sight and maybe into another dimension in space and time. For instance, seeing dream imagery of a horse can mean it has come to fetch someone. If a dreamer sees themselves turning into a horse, it can mean that person is galloping away into another realm. In this scene, the dream imagery of a horse represents a “thin subject” that is able to permeate the barrier between The Living World and The World of the Dead.
Jung said that if you saw a premonitory dream, it does not mean you caused that event. This theory is the opposite of what the Native American Indians believe. If one of them dreams of having done something to another person, s/he is deemed to be guilty of the deed and must make reparations to atone for their “deed”, even though it was only dream imagery seen in the dream. This belief subscribes to Freud’s theory that dreams are the heart’s desires and the dreamer is responsible for wanting to see that dream imagery and hoping it happens in real life. Read more at Guide to Dream Interpretation for Native Americans.
Wikipedia has compiled a list of horses in mythology and folklore.
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Interpret Dream of Riding Horse.