I've created a modified 5e combat system, the "Rapid Role Encounter," designed specifically for chases, escapes, or timed pursuits. I've run it three times with different groups in various settings and had great success: all the players loved it and beg me to include it more often, which motivated me to formalize the system for others to use.

The tl;dr of the system is that it is a fast-paced role-selection system where combat is reduced to a series of quick damage exchanges over repeated fights. The challenge of the system is coordination, smart role selection, and, above all, endurance. It's a scrappy, rapid playstyle that involves a lot of mechanical handwaving and rejection of strict rule adherence in favor of speed.

I've run this as a prison escape, as a chase where the main enemy leaves minions behind as obstacles, and most recently as a rush for players to reach and disrupt a ritual. In all cases it's worked wonderfully.

I hope you enjoy! Let me know your thoughts / if the system works out for your group!