Here's another mechanic we used recently that may be of some value to someone. The PCs were chasing a would-be burglar through the alleyways, and based on the initial encounter this footpad had a lead of 20 feet on them.
They all had the same base movement rate, but PCs were allowed either a Physical (for running extra fast) or Subterfuge (for running extra nimbly) check each round. Success meant gaining 10 feet, failure meant no progress. Any attacks (either by pursuer or pursuee) are at disadvantage, while foregoing your attack allowed you advantage on your "pursuit check." Once any PC got to 0 ft they could attempt to grapple, which these guys did successfully after a couple attempts.
It's a concept I'm sure I convinced myself I made up when in reality I probably read it somewhere in the m20 2017 Collection, or heard it on Not Another D&D Podcast. But it turned out to be pretty fun, especially when the footpad threw a flask of oil behind him (PCs needed a Dex check to avoid it or lose 10 ft of ground as they stumbled to right themselves). It could be really fun if there's also a goal on the other end, e.g. the footpad is running to make a ferry leaving the dock ahead, and the PCs are trying to catch him before he leaps across the water and is out of reach.
Curious to hear thoughts on this versus e.g. using the standard 5E "Dash" action, which IIRC simply doubles your movement rate by sacrificing your primary action.