In round-by-round movement, simply counting off squares, it’s impossible for a slow character to get away from a fast determined character without mitigating circumstances. Likewise, it’s no problem for a fast character to get away from a slower one.
When the speeds of the two concerned characters are equal, there’s a simple way to resolve a chase: If one creature is pursuing another, both are moving at the same speed, and the chase continues for at least a few rounds, have them make opposed Dexterity checks to see who is the faster over those rounds. If the creature being chased wins, it escapes. If the pursuer wins, it catches the fleeing creature.
Sometimes a chase occurs overland and could last all day, with the two sides only occasionally getting glimpses of each other at a distance. In the case of a long chase, an opposed Constitution check made by all parties determines which can keep pace the longest. If the creature being chased rolls the highest, it gets away. If not, the chaser runs down its prey, outlasting it with stamina.
Moving Around In Squares
In general, when the characters aren’t engaged in round-by-round combat, they should be able to move anywhere and in any manner that you can imagine real people could. A 5-foot square, for instance, can hold several characters; they just can’t all fight effectively in that small space. The rules for movement are important for combat, but outside combat, they can impose unnecessary hindrances on character activities.