My house rules are shown in blue. I find that using miniature figures helps when running a chase, so the following rules assume that you are using figures on a grid. Standard combat rules apply except as noted below. Characters that pause to take an action, other than Dash, move a distance equal to their move rate. Most characters use the Dash action and move a distance equal to twice their move rate.
Setup. Determine where everyone involved in the chase is located. The only thing that matters is how far apart everyone is. Place the lead quarry first, then place the others at the appropriate distance behind him. If their locations aren’t pre-determined based on the encounter, you can randomly set the distance from the lead pursuer to the closest quarry at the speed factor of the fastest creature + 5x(1d6) feet.
Determine Initiative. Set initiative order based on position. The lead character is assigned the highest initiative, followed by the others in order of their distance behind him. This initiative order may change from round to round as creatures pass each other. Ties go to the one with the highest dexterity score.
Track Movement. After the lead quarry determines his total move distance – write that distance down so it can be referenced by all players. Don’t move that figure. On each participant’s turn, compare the distance he moved to that of the lead quarry. If they are the same, the distance between them remains the same, so his figure doesn’t move. If he moved farther than the lead quarry, subtract the lead quarry’s move from his and move his figure forward by that amount. If the lead quarry moved farther than he did, subtract his move distance from the lead quarry’s and move his figure back by this amount.
No Opportunity Attacks. No one involved directly in the chase can use an opportunity attack against anyone else in the chase.
Track Exhaustion. You can use the Dash action a number of times equal to 3+ your Constitution modifier. For each Dash action after that you must succeed on a DC 10 Constitution check or take one level of exhaustion. Your speed becomes 0 when you reach level 5.
Pursuer Overtakes Quarry.
Attack. If a pursuer is able to move into a quarry’s space, he may instead use a bonus action to perform a single melee attack against the quarry when he is within reach. The attack is made at a disadvantage. Note that the pursuer cannot use this option if he can only move within reach, but could not overtake the quarry if he chose to.
Overtake. A pursuer overtakes a quarry when he moves into its space. He can then use a bonus action to attempt to grapple the creature. Normal grapple rules apply. If successful, both pursuer and quarry are stopped. Rather than grapple, the pursuer may attempt to trip, push over or tackle the quarry. The pursuer has advantage on the attack. As an optional rule, an attack that fails by 5 or more results in the pursuer falling prone.
Quarry Escapes. The quarry can attempt to escape if it is out of sight for all of the pursuers. He makes a Dexterity (Stealth) check and must beat the passive Wisdom (Perception) scores of the pursuers.
Complications. Roll 1d20 at the end of your turn and compare that roll to the appropriate Chase Complications table. The complication is not applied to your character, but rather to the next character in initiative order. You can spend an inspiration point to negate the complication you rolled or one that affects you. Rather than rolling on the table, the DM may allow a quarry to impose a condition on a pursuer to slow him down. It might be one listed on the table, or one of his own creation. Another option to using a table would be for the DM to declare conditions based on his map or the terrain and the path the quarry takes.