IMO, Pathfinder missed a golden opportunity to marge the Climb, Jump, Swim and Fly skills, and the Run feat into a single Athletics skill. If one of the applications of that skill was to allow characters to increase their speed abovve "double move" (for running), that would automatically provide a simple chase mechanic.
For the most part, I handle chases with a Skill Challenge - each round each character must make a "movement" check (whether it's Athletics, Ride, Drive, or another applicable skill). Any character who fails the check can do nothing in the phase except try to keep up. A character who succeeds on the check can take another action to try to win the chase - find a shortcut (Navigation), get a local to misdirect the opponent (Streetwise), create an obstacle (usually a Str check), etc.
(Obviously, if the party are doing the chasing, then the actions they can take may be modified, but the principle is the same.)
If any of these actions succeed, the party gains a success for the chase. If they all fail, or if the whole party fail their "movement" checks, then they suffer a failure for the chase. And, as usual, they need X successes before 3 failures to 'win'.