'moult' (base of 'moulting') originates in Middle English, ultimately from older Germanic roots referring to change or exchange; the word developed in English to refer specifically to the shedding of an outer covering.

'moult' changed from Middle English forms into the modern English verb 'moult' (and the US form 'molt'), with the sense narrowing over time from a general idea of change to the biological sense of shedding feathers or skin.

Initially, related forms referred more broadly to changing or exchanging; over time the meaning narrowed to the current sense of 'shedding an outer covering'.