c. 1200, flitten, flytten, flutten "convey, move (a thing) from one place to another, take, carry away," also intransitive, "go away, move, migrate," from Old Norse flytja "to remove, bring," from Proto-Germanic *flutjan- "to float," from extended form of PIE root *pleu- "to flow." Intransitive sense "move lightly and swiftly" is from early 15c.; from c. 1500 as "remove from one habitation to another" (originally Northern English and Scottish)
Theire desire ... is to goe to theire newe masters eyther on a Tewsday, or on a Thursday; for ... they say Munday flitte, Neaver sitte. [Henry Best, farming & account book, 1641]
Related: Flitted; flitting. As a noun, "a flitting, a removal," from 1835.
The main modern verbal sense of "play at courtship" (1777) probably developed from the noun (see flirt (n.)) but also could have grown naturally from the 16c. meaning "to flit inconstantly from object to object." To flirt a fan (1660s) was to snap it open or closed with a brisk jerk and was long considered part of the coquette's arsenal, which might have contributed to the sense shift. Or the word could have been influenced from French, where Old French fleureter meant "talk sweet nonsense," also "to touch a thing in passing," diminutive of fleur "flower" (n.) and metaphoric of bees skimming from flower to flower. French flirter "to flirt" is a 19c. borrowing from English.
It forms all or part of: fletcher; fledge; flee; fleet (adj.) "swift;" fleet (n.2) "group of ships under one command;" fleet (v.) "to float, drift; flow, run;" fleeting; flight (n.1) "act of flying;" flight (n.2) "act of fleeing;" flit; float; flood; flotsam; flotilla; flow; flue; flugelhorn; fluster; flutter; fly (v.1) "move through the air with wings;" fly (n.) "winged insect;" fowl; plover; Pluto; plutocracy; pluvial; pneumo-; pneumonia; pneumonic; pulmonary.
It is the hypothetical source of/evidence for its existence is provided by: Sanskrit plavate "navigates, swims;" Greek plynein "to wash," plein "to navigate," ploein "to float, swim," plotos "floating, navigable," pyelos "trough, basin;" Latin plovere "to rain," pluvius "rainy;" Armenian luanam "I wash;" Old English flowan "to flow;" Old Church Slavonic plovo "to flow, navigate;" Lithuanian pilu, pilti "to pour out," plauju, plauti "to swim, rinse."