early 14c., conveien, "to go along with;" late 14c., "to carry, transport;" from Anglo-French conveier, Old French convoiier "to accompany, escort" (Modern French convoyer), from Vulgar Latin *conviare "to accompany on the way," from assimilated form of Latin com "with, together" (see con-) + via "way, road" (from PIE root *wegh- "to go, move, transport in a vehicle").
Meaning "communicate by transmission" is from late 14c. Sense of "act of transferring property from one person to another" is from 1520s. It was a euphemism for "steal" 15c.-17c., which helped broaden its meaning. Related: Conveyed; conveying.
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.
"going along with, adjoining," by 1782, present-participle adjective from accompany (v.).