Skill Challenges, More Descriptors, and Draw Out Time
I view extended situations like running away as extended skill challenges, which were presented in 4e as a means to permit characters to exercise more than just their ability hit a guy.
When it comes to players running away, I would recommend taking a moment as the DM to think about the route they're going to take and work to describe where they are heading in a lot more detail than might be typically necessary. For example, if you simply state that they run through the trees, there's nothing really to work with; instead say, "You sprint through the understory, weaving through the trees in an effort to shake your pursuers your mad dash startles a flock of starlings. Brambles catch on the Wizard's robes, but the Barbarian sees the temporary snag and grabs the Wizard's hand and pulls him loose."
In the first example, as a player, I know that there's trees and have nothing to really work with when the DM asks me, "What do you do?" Probably I'll say, "Um, keep running."
In the second example, as a player, I know that the trees are close enough to require weaving, so maybe I can use them to hide with a Stealth check. There's brambles snagging hard enough to slow people down, can I do something with that maybe with a Knowledge (Nature) or Survival check? There's animals, does the Druid want to consider something like Dominate Animal to make a bird try divebombing at the pursuer?
In general, when coming up with your descriptors, don't do so with how it will solve their current problem. Simply add more flavor to your chase so it's not a chase through a location with only 1 or 2 elements that aren't really impactful to the chase, then let your players use the playground you create to come up with their own ideas.
I think the DMG has charts to come up with various elements that occur during a chase. Don't be shy about using those to come up with random things to add into your chase.
Regarding the passage of time, I don't think it often makes sense for everything to hinge upon a single 6-second round to decide if things succeed or fail. Consider asking for checks every 1 minute of in-game time. This lets powerful, short duration spells be a wing for that round, but not necessarily the entire encounter.
Consider also what the pursuer is doing as well so that these checks can be opposed. For example, if the party's slowest member has a speed of 25' and the pursuer has a speed of 40', the party needs to do things to slow that opponent down in various ways. Perhaps the party's Rogue and Barbarian work together to make a Survival check to push a tree into the path of a pursuing Owlbear, who then needs to make an opposed Acrobatics check to avoid this. On the next round, the Druid casts Dominate Animal on a woodchuck and has it sprint full tilt into the face of the Owlbear whom might need spend a round attacking the woodchuck to kill it and losing ground. On the following round, the Rogue might notice a thick patch of brambles drop some caltrops on the most likely alternate route to be taken, and now the Owlbear needs a Perception check to notice the caltrops.
Your skill monkey Bards and Rogues will probably see more action in these situations, which I consider a feature as it validates the expenditure of their skill points.