FormerlyHemlock
Hero
Yep you cant use standard moves for chasing, it's too predictable. I suggest rolling for random gain/loss distances (I like 3d10), worked well for me.
I like the 2nd edition rules. Modified for 5E:
When moving in a near-straight line (i.e. chase situation), you can use your action to make a special Dash called a Sprint, which comes in multiple levels.
Sprint 1: make a single DC 10 Str (Athletics) check to double your Dash bonus distance this round (so you'd move 90' total instead of 60'). At the end of this round, make a DC 10 Con (Athletics) check to avoid taking one level of fatigue temporarily (goes away next short rest).
Sprint 2: make a DC 10 Str (Athletics) check and a DC 14 Str Athletics (check) to increase your Dash bonus distance this round. One success = double Dash distance, two successes = triple it. At the end of this round, make a DC 14 Con (Athletics) check to avoid taking one level of fatigue temporarily (goes away next short rest).
Sprint 3: make Str (Athletics) checks of DC 10, 14, and 18 to increase Dash bonus distance up to quadruple if all succeed. Con check is DC 18.
Etc., etc.
To sprint as fast as a high school track star (15.9 mph = 139' per round), you'd need to pass all three checks on a Sprint 3 (giving you 150' movement). A Valor Bard might be able to do that pretty reliably. To run as fast as Usain Bolt (28 mph = 246' per round), you'd probably want to start off as a Mobile Wood Elf with 45' of movement and then Sprint 4, passing four checks at DC 10, 14, 18, 22 for a top speed of 270' per round, and then pass a DC 22 check to avoid fatigue.
The net effect is that a monster can hope to put on enough of a burst of speed to catch up and make an opportunity attack or two, but eventually fitness wins out. It makes combat less predictable.