Racing and chasing happens a lot in comics and action movies, but under standard d20 rules, chases are pretty boring. If the racers have the same speed, then there’s no contest. All involved characters take their movement and nobody catches anyone. If the racers have different speeds, then the faster one is guaranteed to eventually catch up. There are factors that can make chases more interesting however, specifically obstacles.
Obstacles include anything that keep the characters from moving their maximum speed , either double-moves or Running, and you use the Climb, Jump, and Tumble skills to overcome those obstacles. The rules do not essentially change, except that the GM will have to know those skills very well and have DCs readied before the game or just get comfortable with estimating them on the fly. If and when GMs set up a chase in a game, or if one just breaks out, they must remember to place lots of interesting things in the racer’s way: cars (both moving and stationary), baby carriages, pets, bike messengers, gates, stairwells, locked doors, plate-glass windows, fire escapes, anything and everything that slows the characters down and makes them use their skills. The more the characters use their skills, the more the chase is decided by those skills and not just by their base speeds. Of course, someone with Super Speed is probably going to catch someone without it, but even a speedster can trip and fall down an open sewer. It’s not the same as trashing the scenery, but remember that the environment is half the fun of any good action sequence in a film, so fill that environment with stuff.
For straight-out chases, the Sprint feat creates the ability to have slightly different running speeds, and therefore stage chases that are potentially quite close, and therefore more exciting. Any players who have characters who use this feat will have to keep track of exactly how many feet in excess of a full square they are, per round. For example, a character who Sprints 38 feet per round needs to keep that floating extra 3 feet in mind and add a square of movement every second round. As long as each individual player (including the GM) keeps track, and they all compare notes, you can tell exactly where people are during a chase, in which a few feet (i.e., arm’s length) can decide the outcome of the scene.
Such encounters can present a practical problem when it comes to mapping. If a chase breaks out between characters who can move fast enough that a standard battlemat will no longer be big enough, it’s best to move to a relative map instead of a fixed map. On a relative map, you place your racers/chasers in positions relative to each other based on their respective speeds and resolve the chase with the assumption they are continuously on the move.
For example, if you have two characters moving at speeds of 120 ft. and 150 ft. respectively, the slower character will be 30 ft. behind at the end of the first round, 60 ft. behind at the end of the second, and so on (assuming both are moving at maximum speed). Alternatively, if the chasers are moving at a Run, then you’ll want to change the map’s scale. Instead of 5-ft. squares, you can move to 10-ft. or even 20-ft.
When speedsters Run, they can easily cover hundreds of clicks over the course of a combat. If you’re math-fu is strong, you can figure the lowest-common denominator of their two speeds and set the map’s scale to that number. For example, if two characters were moving at 360 ft. and 400 ft. per round respectively, you could set a scale of 40-ft. squares because both of their speeds are evenly divisible by 40. On that scale, the faster character could move up to 10 squares, and the slower only 9 squares. Of course, each square still represents 40 feet of space, but now you can depict the chase more easily.
If you have a chase in which the characters move at a Run, then don’t sweat small amounts of lateral movement. Technically, you can Run only in straight lines, but over the course of several hundred feet, a little drift is acceptable, so the chasers might Run along a curved freeway or weave in and around traffic. If it’s just for flavour, this kind of thing can help to make a chase more dramatic. This requires a certain amount of improvisation on the part of the GM and a willingness to rearrange the map on the fly in order to simulate the new environments that your chasers might end up in over the course of the Chase itself. If you run your game in a real city that you know well, you can even estimate how many blocks your characters traverse and the neighbourhoods that they run through, from a downtown core, to outlying residential areas, to suburbs, even out into the hinterlands, and then all the way back.
If the participants have Bonus Actions, then they can of course keep attacking each other or interfering with/saving innocent bystanders the whole time. In fact, that kind of thing is encouraged! The name of the game is danger after all (the name is actually “phoenix,” but you know what we mean). The important thing in this kind of Chase, for the GM, is to keep describing the scenery in detail. Just switching positions relative to each other on the battlemap isn’t nearly as fun as knowing that the wake from your chase just set off a whole block of car alarms, or that you’ve just passed through Edmonton when you started in Seattle.