I've played around with a few ways of handling chases and my most recent iteration abstracts it a bit into the following. I play on the grid for combats, notably. We don't use initiative for chases.
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The Chase Begins
If a creature leaves the area shown by the map and another creature pursues, move to the Chase screen. The quarry is placed in Zone H. The hunter is placed in Zone C.
If the quarry has a faster speed than the hunter, the quarry can automatically escape if they so choose.
Fly, You Fools!
The quarry goes first, moving from its current zone to the next one, but not before encountering a random wilderness complication that may impede progress.
The hunter goes next, moving from its current zone to the next one, encountering the same complication.
A Complication Ensues
The DM presents the complication. The hunter and quarry must describe how it is dealing with the complication and make any relevant ability checks if necessary. If there is more than one hunter or quarry, the creature dealing with the complication cannot have already dealt with a complication (unless all hunters or quarry have done so already). Failing by 5 or more can sometimes carry additional consequences.
Hide!
If the quarry was successful in dealing with the complication, the quarry may attempt to hide, making a Dexterity (Stealth) check against the hunter's passive Perception score. If the hunter failed at dealing with the complication when moving into the current zone, this check is made at advantage.
If the attempt to hide is successful, the quarry escapes and is safely away from the hunter, but cannot help other quarry. If the attempt to hide is not successful, the chase continues (repeat, starting at Fly, You Fools!).
No Place Left to Run
If the quarry fails to hide in or before Section E, the hunter catches up to them and an encounter follows, starting at a range of 1d3 x 30 feet.