Animals on Shabbos
1. [Exodus 23:12] states: "[...cease work on the seventh day] so that your donkey and your ox will rest." Thus, the Torah has commanded us that an animal belonging to a Jew should rest. This commandment applies to all living creatures (1).
Accordingly, it is forbidden to allow one's animal to carry a burden ("Masah") in the public domain (2). Even if it wanders out to the public domain on its own while carrying a burden, the owner is considered to have violated one of the Torah's positive commandments (3).
Even something placed on the animal as an ornament is considered a burden; in contrast, anything placed on the animal for medicinal purposes, for example, a bandage on a wound, [is not considered a burden] and may be worn out [into the public domain by an animal]. Similarly, anything that is necessary to control or guard the animal is considered like a person's clothes and may be worn out by it into the public domain. Any extra or unnecessary safeguard, however, is forbidden. Furthermore, something that is not sufficient to control a particular animal, is considered a burden and is prohibited, even if it is sufficient to control another animal.
(1) That is, even though the verse specifies a donkey and an ox, it is not coming to exclude other types of animals, but just presenting the most common case.
(2) This halacha only applies to one's own animal, not an animal owned by a gentile.
(3) Even if another person placed the burden one's animal, one is obligated to remove it.