Alternate chase rules

When the baker's daughter was revealed to be a doppelganger, the creature jumped out the window and clamored down the lattice like some ridiculous spider, tearing through the market square. It was meant to be thrilling, but the resulting chase scene using the official rules was an arithmetic nightmare for scatter-brained Perkins — probably my most disappointing scene as a DM. I need a more loosey-goosey style that doesn't resemble combat so much. So here I've made an alternate set of rules for chase scenes.

The gist of it

These rules work when multiple pursuers are chasing one quarry. As a modified skill challenge, the goal for the pursuers is to succeed a set number of ability checks before failing three — we're not tracking distances like the official rules. Successes mean the pursuers are gaining ground on the quarry. Failures mean the pursuers are lagging behind. Because the pursuers' performance is relative to their quarry's performance, the DC for each ability check determined by the quarry's roll. There is no initiative! Each round, the quarry makes their ability check, then the pursuers make their ability checks.

Chase order of operations

This example assumes the players are the pursuers chasing an NPC quarry, but the roles could be easily reversed.

1. Set the difficulty of chase. The difficulty of the chase is determined by the number of successes needed to catch the quarry. I would usually stick to three, but you could ramp it up to five and probably still keep it interesting. Regardless, three failures always results in a pursuer dropping out of the chase and suffering a point of exhaustion..
2. Determine advantage or disadvantage based on move speeds. If a pursuing creature has a lower move speed than its quarry, the pursuer rolls every ability check at disadvantage. If higher, the pursuer has advantage. If their speeds are identical, pursuers roll as normal.
3. Choose a complication and appropriate ability check. This can be predetermined or chosen from the lists in the Dungeon Master's Guide. In a jungle, a challenge might be crossing fetid water with a Constitution check. Navigating a crowded tavern might require an Insight check to pick a path through the drunks. Through the desert, you might make an Athletics check to pull your way through quicksand. 
4. The quarry makes a check to determine the DC. I personally wouldn't share their result with the pursuers just yet. Be descriptive about how well the quarry rolled though — this will help clue players in to whether they need to use their inspiration or other abilities.
5. The pursuers make the same ability check, attempting to beat the quarry's DC. Share the target DC after players lock in their rolls.
6. Repeat until the chase ends. A chase ends for everybody when a pursuer reaches the set number of successes. An individual pursuer drops out of contention after three failures.

Special considerations

Making attacks, using abilities, casting spells
Pursuers may use attacks, abilities or class features in lieu of making an ability check, but this will count as a failure for the skill challenge.

Determining range
If a player wants to use a touch effect on an ally, they must both have the same number of successes. If a player wants to use a ranged effect on an ally, they must be within one success of each other. If a player wants to use a ranged effect on the quarry, the pursuer must be on their final ability check (eg. two successes out of three, four successes out of five).

Reducing the quarry's movement
If an effect would stop the quarry from moving (eg. Hold Person, the restrained or grappled condition), every pursuer except the caster of the effect automatically succeeds the ability check. This affects pursuers who have already rolled and failed their check for the round. If an effect would reduce the quarry's movement (eg. Ray of Frost), redetermine advantage or disadvantage based on move speeds as appropriate. In either case, the effect then ends and the chase continues as normal.

Buffing allies
If a player wants to buff an ally, it must be declared before that ally rolls their ability check, unless the effect specifically states it can be used after the roll. The effect then ends and the chase continues as normal.

Bypassing checks
Players are clever, so they will work out ways to bypass some obstacles. That's fine! If a player casts Fly on their ally to bypass the Athletics check and vault over a high wall, the flying character can succeed the check automatically (if you think that's fair). Alternatively, you could give them advantage on the check. Reward creativity in whichever way you think is fair and fun. The effect then ends and the chase continues as normal. 

Combat
Just because the quarry is caught, this doesn't mean they'll always go quietly. If combat begins at the end of the chase, everybody rolls initiative as normal. Characters who have only partially succeeded or dropped out of the skill challenge join the combat on round two.

Three failures
If a character fails three ability checks, they drop out of the chase and suffer a point of exhaustion.

Example of play

Scene: A chase through the busy streets of a city.
Difficulty: Three successes before three failures.
Pursuers: Albert, Bonnie, Cate.
Quarry: Destroyinator.

Determine advantage or disadvantage based on move speeds.
Destroyinator (the quarry) has a movement speed of 30ft.
Albert is slower than the quarry with a movement speed of 25ft, so he rolls ability checks at disadvantage. 
Bonnie is same speed as the quarry with a movement speed of 30ft, so she rolls as normal. 
Cate is faster than the quarry with a movement speed of 35ft, so she rolls ability checks at advantage.

Round 1
Complication: Athletics check to scale a building.
Quarry sets the DC: Destroyinator rolls 10 on its Athletics check.
Albert: Albert is range of her allies. Albert casts Fly on Bonnie in lieu of making the ability check. Albert takes an automatic fail.
Bonnie: Bonnie automatically succeeds because of Albert's Fly spell.
Cate: Cate rolls at advantage because of her movement speed. 15 — success.
Scoreboard: Albert one failure, Bonnie one success, Cate one success.

Round 2
Complication: Perception check to find an unlocked window.
Quarry sets the DC: Destroyinator rolls 15 on its Perception check.
Albert: Albert rolls at disadvantage because of his movement speed. 12 — failure.
Bonnie: 13 — failure.
Cate: Cate rolls at advantage because of her movement speed. 15 — success (meets it, beats it).
Scoreboard: Albert two failures, Bonnie one success, one failure, Cate two successes.

Round 3
Complication: Arcana check to pick a safe path through an alchemist's laboratory.
Quarry sets the DC: Destroyinator rolls 11 on its Arcana check.
Albert: Albert rolls at disadvantage because of his movement speed. 10 — failure.
Bonnie: 12 — success.
Cate: Cate is in range of the quarry. Makes a ranged attack roll with a javelin in lieu of making the ability check. Hits! Deals 8 damage. Cate takes an automatic fail.
Scoreboard: Albert three failures (out of chase, takes a point of exhaustion), Bonnie two successes, one failure, Cate two successes, one failure.

Round 4
Complication: Acrobatics check to leap out a window to street level.
Quarry sets the DC: Destroyinator rolls 19 on its Acrobatics check.
Albert: Out of chase. Does not roll.
Bonnie: Bonnie is in range of the quarry. She casts a movement-inhibiting effect in lieu of making the ability check (Hold Person). It works! Bonnie takes an automatic fail. Everyone else in the chase takes an automatic success. The Hold Person effect only lasts this round.
Cate: Cate takes an automatic success because the quarry has been stopped.
Scoreboard: Albert three failures (out of chase, takes a point of exhaustion), Bonnie two successes, two failures, Cate three successes, one failure. Cate has caught the quarry!

Combat
Everyone roll initiative!
Quarry:
Destroyinator starts combat as normal — it has already taken some damage in the chase.
Albert: Albert did not complete the chase. He joins combat on round two.
Bonnie: Bonnie did not complete the chase. She joins combat on round two.
Cate: Cate starts combat as normal.