oregon
For a horse on the road, Oregon law says: (emphasis added)
Every person riding an animal upon a roadway and every person driving or leading any animal is subject to the provisions of the vehicle code concerning vehicle equipment and operation of vehicles except those provisions which by their very nature can have no application.
“[T]he provisions of the vehicle code concerning [...] operation of vehicles” include the offense of operation of unsafe vehicle:
Drives or moves on any highway any vehicle which is in such unsafe condition as to endanger any person.
This is a class B traffic violation.
Another law that the rider might be cited under is failure to perform duties of person in charge of livestock on roadway.
When riding or leading a horse or other livestock on the highway, a person must keep a lookout for vehicles and use caution to keep the animal under control. [...]
This is a class B traffic violation.
More gravely, if this falls below the standard of “minimum care” of the animal, it would constitute animal neglect in the second degree, and if this led to the injury or death of the horse, animal neglect in the first degree.
“Minimum care” means care sufficient to preserve the health and well-being of an animal and, except for emergencies or circumstances beyond the reasonable control of the owner, includes, but is not limited to, [...]
If this recklessly causes an injury to the animal, it could rise to the level of a criminal misdemeanor:
A person commits the crime of animal abuse in the second degree if, except as otherwise authorized by law, the person intentionally, knowingly or recklessly causes physical injury to an animal.
“Recklessly,” when used with respect to a result or to a circumstance described by a statute defining an offense, means that a person is aware of and consciously disregards a substantial and unjustifiable risk that the result will occur or that the circumstance exists. The risk must be of such nature and degree that disregard thereof constitutes a gross deviation from the standard of care that a reasonable person would observe in the situation.
It might potentially be considered careless or even reckless driving. The latter is a class A misdemeanor.
A person commits the offense of reckless driving if the person recklessly drives a vehicle upon a highway or other premises described in this section in a manner that endangers the safety of persons or property.
Section 811 of the Oregon Revised Statues is part of Title 59, the Vehicle Code referenced above, and thus applies to riding an animal on a roadway.