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late 13c., from Anglo-French mover, Old French movoir "to move, get moving, set out; set in motion; introduce" (Modern French mouvoir), from Latin movere "move, set in motion; remove; disturb" (past participle motus, frequentative motare), from PIE root *meue- "to push away" (cf. Sanskrit kama-muta "moved by love" and probably mivati "pushes, moves;" Lithuanian mauti "push on;" Greek ameusasthai "to surpass," amyno "push away").
Intransitive sense developed in Old French and came thence to English, though it now is rare in French. Meaning "to affect with emotion" is from c.1300; that of "to prompt or impel toward some action" is from late 14c. Sense of "to change one's place of residence" is from 1707. Meaning "to propose (something) in an assembly, etc.," is first attested mid-15c. Related: Moved; moving.
mid-15c., "proposal," from move (v.). From 1650s in the gaming sense. Meaning "act of moving" is from 1827. Phrase on the move "in the process of going from one place to another" is from 1796; get a move on "hurry up" is Americal English colloquial from 1888 (also, and perhaps originally, get a move on you).