So I worked with what we've said so far, and I wrote something up. It incorporates some of what RobbyPants wrote and some of my own ideas. I haven't included SiggyDevil's suggestion because I can't quite figure out what he's getting at (perhaps I'm over-thinking again?). This basically comes down to a new move action, "Sprint," and a change to the Run feat, as well as some guidelines for adjudicating Tumble and Jump checks in combat. (NB: In my games, I've folded Balance into Tumble, but I'll mark the places where you could use Balance.)
Chase Rules
The following rules are designed to work with, and in a couple of circumstances supersede, the standard movement rules for d20.
Starting a Chase is like starting combat. If someone starts to pursue someone or something else, you have a chase on your hands. You can have surprise rounds in Chases just like in combat (someone suddenly comes at your or runs away, without warning), and then everyone roles Initiative and you take turns as normal. You can even run Cases and combats simultaneously, with some people chasing and some fighting. Chasing comes down to two things, Sprinting (a new kind of move action), and Manoeuvrability (which is really just a set of guidelines for adjudicating other movement in Chases).
Sprinting
When you Sprint, you temporarily increase your movement to your Base Speed plus your Dexterity modifier, and you do not lose your Dexterity/ Dodge bonuses to your Defence score. You can Sprint for a number of rounds equal your Constitution modifier +5, after which, at the beginning of every round, you must make a Constitution check (DC 10). The Endurance feat grants a +4 on this check. If you pass the check, you can continue running, but at the beginning of your next round you make a new Con check and the DC increases by +2. If you fail the check, you must reduce your Sprint speed by 5 ft., but the Constitution check goes back down to DC 10, and the whole process starts again. Once you reduce your Sprint speed to 10 ft., you have to rest for 10 rounds (1 minute) before you can Sprint or Run (see the Run feat in Chapter 4 for more Sprinting rules).
Manoeuvrability
Your manoeuvrability comes into play when you have to avoid an obstacle, like jumping over an object, slipping around a gawking bystander, or making a high-speed turn. In such cases, your GM will call for a skill check as determined by the obstacle, usually Tumbling around/through something or someone, or Jumping over it. Unless otherwise stated, if you fail a check, you must reduce your speed a number of feet equal how much you failed by (i.e., the difference between your modified roll and the DC). This reduction applies only to the current round. If it causes you to run out of movement for the round, then you stop where you are and have to start fresh the next round. Unless otherwise stated, if you fail a check by 10 or more, you fall prone, with all the logical consequences of doing so, including falling between two buildings or stumbling into traffic, for example. [See table, attached.]
Jump
You can Jump over any obstacle that your Jump score would allow, including people or open spaces (the most common being the chasm between two rooftops). Remember that when you make a running long Jump, your distance equals your modified Jump roll in feat. You can also make a running high Jump, in which case the distance equals your modified Jump roll in feat, divided by 4. If you fail to Jump over an open space, you can fall, as per the standard Jump rules.
Avoid People/Objects
You can Tumble around an obstacle, as if you were trying to move around an opponent in a football game, or you can attempt to Tumble through their space. Avoiding anything smaller than small-sized does not reduce your speed.
Turn Corners
When you Sprint, you have a limited ability to turn corners (as opposed to a Run, in which you can't turn corners at all). Reckon the angles based on whatever is closest. [Use the Balance skill for this.]
Uneven Ground
Anything other than flat, relatively solid ground can also slow you down. Pavement or firm grass do not slow you, but mud/muddy grass, sand, or rocks do. Uneven Ground is not the same as Difficult Terrain, which simply halves your movement.
Run
[General]
You are fleet of foot.
Benefit: When you perform a Run action, you multiply your speed by 5 instead of 4 (or 4 instead of 3, if you're carrying a heavy load or wearing heavy armour). If you make a Jump check with a running start (either taking a move action, Sprinting, or Running), then you get a +4 to any Jump checks you make as part of that movement. While you Run, you retain your Dexterity bonus to your Defence score. When you Sprint, you increase your speed by double your Dexterity modifier.
Normal: When you Run, you move four times your speed (three times if you're carrying a heavy load or wearing heavy armour) and you lose your Dexterity bonus to Defence. When you Sprint, you increase your speed by a number of feet equal to your Dexterity modifier.