early 14c., "move, go, go quickly; move (something) along, carry," probably from a Scandinavian source akin to Old Norse sveigja "to bend, swing, give way," Old Danish svegja, perhaps merged with an unrecorded Old English cognate. The whole group might be related to swag (v.) and swing (v.). The sense of "swing, waver, move in a swaying or sweeping motion" is from late 14c. Meaning "move from side to side" is from c. 1500; transitive sense "cause to move from side to side" is from 1550s (according to OED, not common before 19c.). Figurative sense "cause to be directed toward one side, prejudice" is from 1590s. Related: Swayed; swaying.
c. 1300, "movement from side to side," from sway (v.). The meaning "controlling influence" (as in to be under the sway of) is from 1510s, from a transitive sense of the verb in Dutch and other languages.