early 14c., from Old French poulser (Modern French pousser), from Latin pulsare "to beat, strike, push," frequentative of pellere (past participle pulsus) "to push, drive, beat" (see pulse (n.1)). Meaning "promote" is from 1714; meaning "approach a certain age" is from 1937. For palatization of -s-, OED compares brush (n.1); quash. Related: Pushed; pushing.
To push (someone) around is from 1923. To push (one's) luck is from 1754. To push the envelope in figurative sense is late 1980s. To push up daisies "be dead and buried" is from World War I.