Fauchard1520
Adventurer
The best chase scenes I've seen in D&D were mid-4e Skill Challenges, that incorporated both individual and group checks. 5e has retained group checks, so you could lift an SC structure directly if you're already familiar.
If not, you could use the DMG system, at least as a starting point.
Or you could just stick to the play loop, the bad guy runs for it, the PCs declare the actions they'll use to try to catch him, you narrate the results, calling for checks where there's uncertainty.
If part of the point is to get the whole party caught up with the quarry so they can capture/kill/defeat it, then group checks could still be a useful tool, in that, based on the terrain and situation. Athletics for a flat-out sprint, Streetwise for following through narrow alleys without much line of sight to the quarry, acrobatics for a precarious environment and/or one with many obstacles like swinging through a jungle canopy or navigating a crowded marketplace or running on rooftops, endurance to exhaust and run down a faster quarry, etc.
Is there any place to check out a 4e Skill Challenge style race done properly? Maybe an especially good module that features 'em?