c. 1300, dragge, "dragnet," perhaps from a Scandinavian source (compare Old Norse dragga "a load," Swedish dragg "grapnel") or from Old English dræge "dragnet," related to dragan "to draw" (see drag (v.)).
From 1708 as "anything attached to a moving body that retards its progress." As the name of a device for retarding or stopping the rotation of wheels, 1795. Sense of "annoying, boring person or thing" is 1813, perhaps from the mechanical senses or the notion of something that must be dragged as an impediment.
Sense of "women's clothing worn by a man" is by 1870, perhaps originally theater slang, from the sensation of long skirts trailing on the floor (another guess is Yiddish trogn "to wear," from German tragen); drag queen "male transvestite or cross-dresser" is from 1941.
Drag racing (1947), is said to be from thieves' slang drag "automobile" (1935), perhaps ultimately from slang sense of "wagon, buggy" (1755), because a horse would drag it. By 1851 this was transferred to "street," as in the phrase main drag (which some propose as the source of the racing sense).
In addition to the time trials there are a number of "drag races" between two or more cars. They are run, not for record, but to satisfy the desire of most Americans to see who can get from here to there in the fastest time. ["Popular Mechanics," January 1947]