Etymology
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molt (v.)

also moult, c. 1400, mouten, of feathers, hair, etc., "to be shed, fall out," from Old English *mutian "to change" (in bemutian "to exchange"), from Latin mutare "to change" (from PIE root *mei- (1) "to change"). Transitive sense, "to shed or cast (feathers, fur, skin)" is by mid-15c. With unetymological -l-, late 16c., on model of fault, etc. Related: Molted, moulted; molting, moulting. As a noun, "act or process of shedding an outer structure or appendage," from 1815.

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*mei- (1)
Proto-Indo-European root meaning "to change, go, move," "with derivatives referring to the exchange of goods and services within a society as regulated by custom or law" [Watkins].

It forms all or part of: amiss; amoeba; azimuth; common; commune; communicate; communication; communism; commute; congee; demean; emigrate; emigration; excommunicate; excommunication; immune; immutable; incommunicado; mad; mean (adj.1) "low-quality;" mew (n.2) "cage;" mews; migrate; migration; mis- (1) "bad, wrong;" mistake; Mithras; molt; Mstislav; municipal; munificent; mutable; mutant; mutate; mutation; mutatis mutandis; mutual; permeable; permeate; permutation; permute; remunerate; remuneration; transmutation; transmute; zenith.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit methati "changes, alternates, joins, meets;" Avestan mitho "perverted, false;" Hittite mutai- "be changed into;" Latin mutare "to change," meare "to go, pass," migrare "to move from one place to another," mutuus "done in exchange;" Old Church Slavonic mite "alternately;" Czech mijim "to go by, pass by," Polish mijać "avoid;" Gothic maidjan "to change."
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mew (n.2)

"cage for birds; place where hawks are put to molt," late 14c., from Old French mue "cage for hawks," especially when molting, from muer "to molt," from Latin mutare "to change" (from PIE root *mei- (1) "to change"). In extended use, "a place of retirement or confinement" (early 15c.). Also as a verb, "to shut up, confine" (mid-15c.).

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molten (adj.)

"melted, in a state of solution," c. 1300, from archaic strong past participle of Old English meltian, a class III strong verb (see melt (v.)).

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mutton (n.)

"flesh of sheep used as food," c. 1300, mouton (c. 1200 as a surname), from Old French moton "mutton; ram, wether, sheep" (12c., Modern French mouton), from Medieval Latin multonem (8c.), probably [OED] from Gallo-Roman *multo-s, accusative of Celtic *multo "sheep" (source also of Old Irish molt "wether," Mid-Breton mout, Welsh mollt), which is perhaps from PIE root *mel- (1) "soft."

The same word also was borrowed into Italian as montone "a sheep," and mutton in Middle English also could mean "a sheep" (early 14c.). Transferred slang sense of "food for lust, loose women, prostitutes" (1510s) led to extensive British slang uses down to the present day for woman variously regarded as seeking lovers or as lust objects. Mutton chop "cut of mutton (usually containing a rib) for cooking" is from 1720; as a style of side whiskers from 1865, so called for the shape (narrow and prolonged at one end and rounded at the other).

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