The Survival skill can be used without much issue. You require to make a single check, and if your quarry is moving at twice their speed, or even normal speed, they are not taking the necessary procedures to hide their track (ie: Stealth, which reduces your movement by half), and can be followed for as long as you want without making additional checks once you have found their track.
An animal companion with the Tracker archetype can move their normal speed at no penalty (at 6th level), and twice their speed with a -10 penalty when tracking by scent. A common dog has a Survival bonus of +8, meaning that they are a cheap solution and you could have many of them helping each other (see Aid Another) for a total of +8, plus +2 per additional dog.
If the dogs get a fresh trail, the DC for their Survival check is only 10, +2 per additional hour since the trail was made. So even if they are hiding their tracks (+5 to the DC and move at half speed), the dogs can casually Take-10 to keep tracking their quarry.
Keep in mind that a creature cannot normally move at twice their movement speed for long distances without becoming fatigued, and eventually exhausted. They can move twice their movement speed for an hour by hustling without getting fatigued, but once they hustle for another hour, they become fatigued, and if they do so again, they become exhausted. Once exhausted, they are done for, as the penalties will be enough to make them move at half their speed from now on. Which will make it even harder to hide their tracks, as using stealth will make them move at 1/4 their speed.
The nonlethal damage caused by forcing a march will prevent the character from recovering those conditions until the damage is healed.
In other words, unless they take a standard pace, they are bound to get tired and be forced to rest, so you will eventually find them. If they stop and try to hide their tracks, then you have time to get closer. But if you got a high enough bonus on your check, you will find their track again and keep pursuit. Slowly but certain.
If your goal is to have a high enough bonus so you can always track whoever you want, a ranger is probably the best option, considering that they gain a flat +1 bonus per 1/2 level, an increasing Favoured Enemy bonus, and another bonus if this pursuit is on their favoured terrain. So, at 3rd level we are looking at +5 bonus on top of whatever skill ranks, traits, feats or ability score you got.
At higher levels, this ceases to be a problem, as you have abilities like Swift Tracker (8th for rangers, 11th for slayers). But at levels that high, characters have non-mundane ways to make themselves harder to track (haste, fly, blink, planeshift, teleport, etc), from magical abilities or magic items.
You are describing a Pursuit
However. There are (better, personally) optional rules for long distance Pursuits in the Ultimate Intrigue, which basically divides the pursuit into 1-hour long phases where the pursuers must make survival checks and compare their results with their quarry's total movement in that 1-hour period.
A quarry can take the Obscure Trail tactic to attempt to make the pursuers lose your track:
Obscure Trail: Mark the terrain tile where a character starts and stops using this tactic. A character using this tactic reduces her progress by half in order to increase the DC to track her group by 5 throughout the marked section. This tactic requires the group to be quarries.
While the pursuers have the option to Fast Track:
Fast Track: A character using this tactic does not reduce her progress by half while tracking. However, she takes a –5 penalty on the Survival check to track. Abilities such as the ranger’s master hunter class feature negate this penalty.
So, using this optional rule, each group is trying to get advantage over the other while crossing a 12-mile long terrain, which works better than simply comparing movement speeds as if they were all walking in a straight line.