Granted, this might not be everyone's experience, but these are the solutions that make the most obvious use of the rules of the more recent editions of D&D. I'm sure plenty of GM's ignore the square counting and go for a more skill-challenge type solution. But skill challenges (as presented in 4E D&D) have their own set of problems, namely that the strict number of failures vs number of successes doesn't necessarily match the narrative description of the scene. If a skill should end the challenge even if the requisite number of successes hasn't been reached, the GM will usually either fudge it and say that the players succeed, or you'll get frustrated players who don't feel like their specific actions mean anything more than "checking a box."
I know I never really had many chases in my games, but recently I've been sprinkling them in. In general, I've been using a modified version of the Thriller Chase rules of the Gumshoe game "Night's Black Agents." The framework works really well for virtually any system, ignoring of course the Gumshoe-specific rules. The result is a narrative, fast-paced system for resolving chases with skill checks.
- First, create a chase track. I like to have the PCs start at a value of 0, and track lead by going up or down on the track. A good default is to cap the track at +/-5. If one or more PC gets to 5, they win. If all PCs get to -5, they lose. This works regardless of whether they're pursuing or pursued (just inject whatever narrative description fits the scene).
- Most importantly, describe a dynamic environment and let the PCs react to it. Picture each round as a leg of the chase that takes place in a discrete scene. Running through a busy city street, a pursuing NPC might make a check to convince guards/police (or upstanding citizens) to stop the PCs. The key here is to not let the chase devolve into "I sprint," "ok, roll Athletics," etc. That's the equivalent of fighting generic orcs in a featureless 30x30 room. As a GM, you'll probably have to play the NPC as a proactive party, actively creating problems/opposition that the PCs are forced to react to.
- Each "leg" or point of lead is a narrative unit; it doesn't have to equal a specific number of feet or squares. The chase is broken up into units of "interesting scenes," and each scene should be summarized with a single skill check for all parties involved.
- As a baseline, a successful skill check will bring the PC up 1 step on the chase track. A failure will drop them down 1 step. Conversely, a success by the NPC (or group of NPCs) might bring all PCs down 1 step, and a failure bring all PCs up 1 step. Depending on the narrative, multiple NPCs might only effect one or some PCs.
- If the difficulty of a check is going to be harder than typical (i.e. the action is riskier), reward that by letting a success increase the lead by 2. Don't get too generous with this option, and typically only use it if there is an easier alternative and the contrast between the options is noticeable. A failure doesn't necessarily have to drop the PC down by 2, though. Equally interesting is a penalty to their next skill check, or even a penalty to an ally's check. Feel free to adjust the "pacing" of the track if the chase stalls or seems to look like it's going to be a back-and-forth for a long time.
- The option of increasing/decreasing lead by multiple increments could also be used if the system has a "degrees of success" mechanic, even if it's as simple as a "crit" (natural 20) in a d20 system. Things like succeeding with style in Fate, great/extraordinary successes in TOR, or getting a raise in Savage Worlds would apply here.
- If the system uses a resource-management mechanic (like Vancian spells), feel free to grant an auto-success for the expenditure of resources, or at least force the NPC to make a difficult skill check that can benefit the party (will increase their position), but won't set them back if the NPC succeeds.
So give it a try next time you have to run a chase scene. If it's completely on-the-fly, consider dropping the chase track down to +/-4, or even 3, because that will keep the number of scenes that you'll need to come up with to a minimum. Note that a +/-3 could be resolved in as few as 2 "rounds," but if there's more back and forth it might take longer. If you need to adjust the pace, make the next scene harder if the PCs are doing better that you anticipated, or easier if they're floundering (especially the latter!). If it starts to get repetitive or you run out of scenes, better to end the chase in an exciting way (some sort of complication?) than slog through the chase track just for the sake of finishing it. This isn't meant to be a hard-and-fast resolution system; it's more of a tool for the GM to ballpark their pacing.