late 13c., "follow with hostile intent, follow with a view of overtaking," from Anglo-French pursuer and directly from Old French poursuir (Modern French poursuivre), variant of porsivre "to chase, pursue, follow; continue, carry on," from Vulgar Latin *prosequare, from Latin prosequi "follow, accompany, attend; follow after, escort; follow up, pursue," from pro- "forward" (see pro-) + sequi "follow" (from PIE root *sekw- (1) "to follow").
The meaning "to proceed, to follow" (a path, etc.), usually figurative (in reference to a course of action, etc.), is from late 14c. This sense also was in Latin. The meaning "seek, seek to obtain" also is late 14c. Related: Pursued; pursuing. For sense, compare prosecute.