Probably a variant of mash, or from a dialectal variant of Middle English mos; compare Middle English appelmos ("applesauce"), from Old English mōs ("food, victuals, porridge, mush"), from Proto-West Germanic *mōs, from Proto-Germanic *mōsą ("porridge, food"), from Proto-Indo-European *meh₂d-. Cognate with Scots moosh ("mush"), Dutch moes ("pulp, mush, porridge"), German Mus ("jam, puree, mush"), Swedish mos ("pulp, mash, mush"). See also moose.
From Old High German muos and mus or muss, or any thick preparation of fruit.
Believed to be a contraction of mush on, from Michif, in turn a corruption of French marchons! and marche!, the cry of the voyageurs and coureurs de bois to their dogs.