When someone is described as pursuing, they are following or chasing another person (or thing) with the intent to catch. The pursuing player in a game of “tag” is “it,” or the one doing the chasing.
During a traditional English fox hunt, the barking dogs are the pursing animals, and the foxes are the ones being hunted. And during a typical cops-and-robbers chase, the pursuing officers are the ones in the police car with the siren blaring, racing after the getaway car. Pursuing and its related verb pursue come from the Latin prosequi, “follow or follow after.”
following in order to overtake or capture or as accompaniment to such pursuit“the fox fled from the
pursuing hounds”“listened for the hounds’
followinggoing or proceeding or coming after in the same direction