I just looked at the DMG and it seems pretty straight forward to me how to transition from combat to a chase. One group flees (becoming a the quarry) and the other chases (becoming the pursuer). What else do you need?
It's not so hard.
Players: We run away!
DM: Ok, we'll switch to the chase rules.
(The real problem is that the chase rules are less than great, but at least they exist)
It's fascinating to me how often people point to the DMG's chase rules when it's painfully obvious that they've never used them, or even thought very much about how it might go down if they tried. What you're saying might work out just fine if you were playing with side initiative and both teams lined up across from each other like a Final Fantasy game. But that's not 5e. So at what point in the round are you allowed to do this conversion from combat to chase? Any time you like? Can you take all the PC turns attacking, then flee before the enemy gets a turn? Do you get to Dash away, then call for retreat and get to start with a free extra 60ft head start? If you're completely surrounded, can you still declare a flee action and teleport through enemy spaces? If you're mostly surrounded, can you declare a flee action to avoid taking half a dozen opportunity attacks? Or do you still take the opportunity attacks anyway? If you do, what happens if the opportunity attack has the ability to stop movement, like sentinel feat or a monster attack that grapples on hit? What about Warcaster and Booming Blade? If you declare that you're fleeing, and the enemy pops you with Booming Blade, are you committed to the chase and automatically take the damage, or can you abort it?
Most hilariously, if you start from melee range, then the distance between the pursuer and the quarry is zero. Congratulations, you've now caught the quarry and the chase is over.
Yes, they exist, but they're not just less than great. They're absolute garbage, and whoever wrote them also literally never considered how to transition from a combat to a chase during and actual game. And, on top of all that, they still mostly come down to "whoever has the highest movement rate wins." And I know someone is about to say something about the stealth checks to escape, but you don't get to make those unless you can break line of sight, and you can't break line of sight unless you can create some distance from the pursuer, and you can't create that distance unless you either started with it, or you're faster than whatever is chasing you. Oh look, that's a whole page of rules that says "the rogue can use Cunning action to get away, then Stealth Expertise to hide. Everyone else is either a crap shoot, or has basically zero chance to escape." You know... the same thing you get if you just don't use the chase rules at all.
