Every creature has a basic movement rating - people with higher movement speeds are faster than those with lower movement speeds. For most intents and purposes, this works just fine. But when you are pursuing an enemy, especially if you are similarly fast or if you are pursuing an enemy over a long distance, more fine-grained rules are needed.
All three of the physical attributes can play a role in pursuit. Strength allows a short-term burst of speed to be put on to gain on a nearby quarry. Constitution allows you to run fast and far, making it possible to wear out a distant quarry in the long run. And Dexterity is useful in cities, forests and anywhere else where you might try to evade capture by misleading and confusing your pursuer.
When adjudicating a chase, the first thing to do is figure out the base speed of each participant. Humans, with a movement rating of 12, can sprint at 60 times that value - 720 feet per round. If either the quarry or the pursuers are much faster than the others, then it's probably not worth keeping track of small gains and losses. If a halfling is fleeing a human, a gain of 5 feet isn't going to make much of a difference when the halfling moves 360 feet per round and the human moves 720.
If their speed is relatively evenly matched, however, you can use the following rules. Every round, the quarry makes a Strength check. For every point they pass by, they get a 1-foot lead. If they fail, it's the same as if they passed by 0. Each of the pursuers also makes a Strength check, which is subtracted from the quarry's lead.
For example, a fighter is chasing a thief, with a 30 foot headstart, who just picked his pocket. The thief makes a Strength check and passes by 3, giving him a 3-foot lead. However, the fighter passes his Strength check by 5, meaning that he actually gains 2 feet. Now the thief is only 28 feet ahead of him.
Neither Strength check can apply "negative" modifiers - if either party fails their check, it's the same as passing by 0. This means that if both pursuer and quarry fail, the lead neither grows nor shrinks. A character with the Running NWP adds 1d4 feet to his lead each round he passes the Strength check.
The rules above are all well and good for short chases of less than 5 rounds. After 5 rounds, however, exhaustion starts to set in. On the 6th round of pursuit, each participant must make a Constitution check. On every subsequent round, they must make another Constitution check with a cumulative penalty each time: -1 on the 7th round, -2 on the 8th, and so on.
The first time they fail a Constitution check, they become fatigued. This imposes a -2 penalty on all dice rolls, including subsequent ability score checks. They may continue running, but must continue to make Constitution checks (the cumulative penalty resets). The second time they fail a Constitution check, they become exhausted. They collapse and cannot run any more; they take a -5 penalty on all dice rolls and can only move at their normal speed (120 feet for a human).
Characters with the Endurance NWP do not have make their first Constitution check until 10 rounds have passed.
Even if you're not strong enough to outrun your pursuers, it still may be possible for a cunning person to evade them by some other means.
The mechanism for adjudicating this is simple. The quarry leads the chase, since the pursuers are just following and trying to keep up with them. Any time the quarry makes use of an opportunity to throw off their pursuers, all participants in the chase dice for initiative. If the quarry loses initiative, they fail. Otherwise, they succeed at what they were attempting - with results to be determined by the DM.
To take the example of the fighter and thief from earlier: the fighter is chasing the thief, when the thief suddenly turns left. The fighter and thief dice for initiaitve. If the fighter wins, he gets to the corner in time to see the thief disappear into the alleyway on the left. If the thief wins, he ducks into the left alleyway before the fighter gets to the corner - and the fighter has to guess which way he went.
Another, higher-risk example. The thief is fleeing the fighter, who is hot on his heels. Knowing he'll be caught if he doesn't do something, he takes a running leap at a wall that the fighter is too slow and heavy for and attempts to climb it. They dice for initiative. If the thief wins, he reaches the wall and can make a Climb Walls check to get over it. If he fails, the fighter reaches him and can attempt to attack him or grapple him to stop him climbing it.
Even though sprinting normally prevents you from attacking in your round, there is an exception for pursuit. If you and your quarry are both sprinting, one pursuing the other, and you catch up - you are automatically considered to be engaged as long as you are armed. If they then flee, you are entitled to an opportunity attack.