If you can catch up to the character, it should. As soon as the pursuer stops moving, they will engage the enemy (meaning the enemy stops). The enemy can move away, but that immediately provokes a Disengagement Attack.
Does that only apply to a player character pursuing an enemy NPC? Or will it apply vice versa - that if an enemy pursues one of my player characters and succeeds in Engagement, will my character immediately stop movement, or will my character continue to move and suffer a Disengagement Attack?
I would prefer that if an enemy Engages me while I'm fleeing, then beginning Engagement stops my movement. If I still want to continue movement and voluntarily suffer a Disengagement Attack, I can choose to do so afterwards. Basically, this would give me a "decision point" the same way it gives AI a clear decision point.
Otherwise, it seems that noticing Engagement and then avoiding the subsequent Disengagement Attack would be an extremely short time window, resulting in a too much reflex based gameplay. On another forum, someone suggested that there could be an Auto-Pause on Engagement. That is an objectively good solution, but subjectively I predict myself to find it clunky. Or perhaps Movement Stop on Engagement could be a toggleable gameplay option.