"kind of porridge; meal boiled in water or milk until it forms a thick, soft mass," 1670s, in the American colonies, a variant of mash (n.) "soft mixture." Meaning "anything soft and thick" is attested from 1824.
command to sled dogs, 1897, first recorded 1862, as mouche, perhaps altered from French marchons! "advance!" (imperative of marcher "to march;" see march (v.)). Related: Musher.