1866, of persons, "to have the health or strength reduced," from the verbal phrase; see run (v.) + down (adv.). From 1896 of places, "dilapidated, shabby, seedy;" by 1894 of clocks, etc., "completely unwound." The earliest sense is "oppressed" (1680s). Compare rundown (n.).
The verbal phrase run down as "have the motive power exhausted" (of clocks, etc.) is by 1761; of persons, etc., "become weak or exhausted," by 1828. To run (something or someone) down "disparage, abuse" is by 1660s.
to win something hands down (1855) is from horse racing, from a jockey's gesture of letting the reins go loose in an easy victory.
The Two Thousand Guinea Stakes was not the best contested one that it has been our fortune to assist at. ... [T]hey were won by Meteor, with Scott for his rider; who went by the post with his hands down, the easiest of all easy half-lengths. Wiseacre certainly did the best in his power to spoil his position, and Misdeal was at one time a little vexatious. [The Sportsman, report from April 26, 1840]
Ancient Greek had akoniti "without a struggle, easily," from akonitos (adj.), literally "without dust," specifically "without the dust of the arena."