to chase after someone who is fleeing, especially a beaten enemy in war
This is the word's home base: a fast, physical chase after a person or group running away. It is the verb for an army breaking the enemy's line and running them down, the verb for a man tearing off after thieves who have raided his camp, the verb for a victor pressing the advantage so the rout becomes a slaughter.
The scenes are concrete and repeatable. Israel pursues the Canaanites down the pass; Pharaoh's chariots pursue Israel to the sea; David asks whether to chase the raiding band. The point of the chase is almost always to catch and finish, so the word often pairs with striking down, overtaking, or destroying what it runs after.
It does not by itself say who is right. The same verb covers a just pursuit of attackers and a wicked king's pursuit of an innocent man. What it fixes is the picture: one side runs, the other runs after.