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.