late 14c., "persecution" (a sense now obsolete), also "a chase with hostile intent," from Anglo-French purseute, pursuite, Old French porsuite "a search, pursuit" (14c., Modern French poursuite), from porsivre (see pursue).
Meaning "action of following briskly for the purpose of overtaking" (regardless of intent) is from mid-15c. Sense of "one's profession, recreation, etc." is attested from 1520s on the notion of "object of one's continued exertions, what one follows or engages in." As a type of track cycling race from 1938.