c. 1300, pushen, "to shove, move onward, strike with a thrusting motion, thrust forcibly against for the purpose of impelling," from Old French poulser (Modern French pousser), from Latin pulsare "to beat, strike, push," frequentative of pellere (past participle pulsus) "to push, drive, beat" (from PIE root *pel- (5) "to thrust, strike, drive").
Transitive meaning "urge, incite, press" is by 1570s; that of "promote, advance or extend by persistence or diligent effort" is from 1714; intransitive sense of "make one's way with force and persistence (against obstacles, etc.)" is by 1718. The 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 "bully, browbeat, domineer" is by 1923. To push (one's) luck is from 1754. To push the envelope in the figurative sense is by late 1980s.