21. [With regard to animals and fowl] that have been trained and domesticated, and usually return to their places in the evening: if they go out on Shabbos and one fears that they will be stolen, one is permitted to drive them into a safe place. One may not, however, carry them by hand, because they are "muktzah" (1).
(1) There is a rabbinical prohibition against riding or even making use of an animal's body on Shabbos or Yom Tov (Beizta 19a). The reason given by the Talmud Bavli is a concern that one may break off a tree branch (a biblical violation of Shabbos) in order to strike it while leading it along or riding it. According to the Jerusalem Talmud, the reason for the rabbinical prohibition is a concern that one may come to lead the animal into a public domain with a load on its back, which we saw is Biblically prohibited (Aruch Hashulchan 305:16). Due to this rabbinical prohibition against making use of an animal, it is considered "Muktzah," which means one cannot move it or pick it up.
There is a dispute among the authorities as to whether one may pick up an animal by hand if it is the only way to prevent it from escaping or being damaged. (See Mishna Berura 305:70, and 308:151).