Chases

Chases are run with slightly different rules that typical combat. The following is quoted from the DMG (pg. 252)

Beginning a Chase

A chase requires a quarry and at least one pursuer. Any participants not already in initiative order must roll initiative. As in combat, each participant in the chase can take one action and move on its turn. The chase ends when one side drops out or the quarry escapes.

Running the Chase

Participants in the chase are strongly motivated to use the Dash action every round. Pursuers who stop to cast spells and make attacks run the risk of losing their quarry, and a quarry that does so is likely to be caught.

Dashing

During the chase, a participant can freely use the Dash action a number of times equal to 3 + its Constitution modifier. Each additional Dash action it takes during the chase requires the creature to succeed on a DC 10 Constitution check at the end of its turn or gain one level of exhaustion.

A participant drops out of the chase if its exhaustion reaches level 5, since its speed becomes 0. A creature can remove the levels of exhaustion it gained during the chase by finishing a short or long rest.

Spells and Attacks

A chase participant can make attacks and cast spells against other creatures within range. Apply the normal rules for cover, terrain, and so on to the attacks and spells.

Chase participants can't normally make opportunity attacks against each other, since they are all assumed to be moving in the same direction at the same time.

However, participants can still be the targets of opportunity attacks from creatures not participating in the chase. For example, adventurers who chase a thief past a gang of thugs in an alley might provoke opportunity attacks from the thugs.

Ending a Chase

A chase ends when one side or the other stops, when the quarry escapes, or when the pursuers are close enough to their quarry to catch it.

If neither side gives up the chase, the quarry makes a Dexterity (Stealth) check at the end of each round, after every participant in the chase has taken its turn. The result is compared to the passive Wisdom (Perception) scores of the pursuers. If the quarry consists of multiple creatures, they all make the check.

If the quarry is never out of the lead pursuer's sight, the check fails automatically. Otherwise, if the result of the quarry's check is greater than the highest passive score, that quarry escapes. If not, the chase continues for another round. The quarry gains advantage or disadvantage on its check based on prevailing circumstances, at the DM's discretion.

Notes

Escape doesn't necessarily mean the quarry has outpaced its pursuers. For example, in an urban setting, escape might mean the quarry ducked into a crowd or slipped around a corner, leaving no clue as to where it went.

Also, there may be complications during the chase.