Once again the lack of skill challenge shows. Though I think there needs to be added some kind of meta currency to make check more reliable or to buy an autosuccess.
For example the fighter and the cleric are being chased and run across the bazaar where they see a crowd of beggars. The cleric throws some coin in front of the beggars and the reverse their cloaks and pretend to be beggars scrambling for coin. Spending some meta plot coupon to buy a success. May be at the price of giving the DM a future use of a complication depending on the die roll.
That's why I use the chase rules as inspiration. You don't need a meta plot coupon, you just need some way of adjudicating whether the ruse worked. In this case it's probably a contest and not guaranteed to succeed.
One thing I do is have a cheat sheet of what PCs are good at, which I can easily see because I use DndBeyond. So let's say the chase is on a busy street during market/rush hours. The PCs are chasing after Timmy Two-Fingers. Timmy scares some horses on purpose by using an acrobatics to dive under the horse which freaks it out. I'll say Timmy rolls high, the horses freak and the cart overturns, barrels go rolling everywhere.
The PCs trailing behind now have to overcome this obstacle. They could try their own acrobatics check to slide under the horses, they could use athletics to just bash through the barrels, investigation to calculate how the barrels are moving. Maybe they have boots of springing and striding but I'll ask for a perception check to see if they notice the fruit cart on the other side. Scenes will, of course, vary.
When stuff like this happens I'll describe the scene and then, based on the PC's best attributes give them an idea or two of what they can attempt. They can always try something else of course. I do run into people now and then that want to use board game rules and cast magic missile or whatever, but I just explain that they only see the fleeing target briefly or are literally just following the chaos the pursued is creating.
When it comes to speed, I just explain it like a car chase, if car chases actually happened on crowded streets the top speed of the cars doesn't really matter. You
can't run full out most of the time, you're constantly dodging or pushing people aside, assessing alternate routes and possible shortcuts.
Of course the scenario has to be right for this kind of scene, but it's not hard to come up with options with practice.