"to move heavily or unsteadily," 1520s, probably from a Scandinavian source akin to Old Norse sveggja "to swing, sway," from the same source as Old English swingan "to swing" (see swing (v.)). Related: Swagged; swagging.
Colloquial sense of "promotional material" (from recording companies, etc.) was in use by 2001; swag was English criminal's slang for "quantity of stolen property, loot" from c. 1839. This might be related to earlier senses of "round bag" (c. 1300) and "big, blustering fellow" (1580s), which may represent separate borrowings from the Scandinavian source. "The primary meaning was 'a bulging bag'" [Klein].