As a starter, correct me if I'm wrong, but you're discussing This Quasit?
Unfortunately, D&D does not have a graceful mechanic for chase scenes. Essentially I would leave the game in whatever state it is in between combat and narrative and decide based on that. To accurately convey where the creature is, the Oracle needs to know the allies' positions in relation to (let's say) him. The characters don't see a battle map with everyone's positions all nice and pretty. They can't see each other either. Oracle has to say "[Summoner], send them fifteen feet forward, five feet left from you" and that barely stays a free action for speech in most circles. In theory the eagles only wouldn't run into each other because they move as separate actions in combat mode.
The initiative order stays and that's what makes things dangerous. The Oracle can speak as a free action, but unless the quasits do, he has to wait until the second one moves to call out directions. Now, while the Oracle can see, this really only narrows down the field for what they can direct. If a quasit moves after the Oracle's turn, they technically have to wait until their next turn before dictating directions. It gets extremely messy because the quasits are technically holding their action until they can see the eagles. The eagles can pretty much be pointed in a direction, and technically every square they move it can be dictated what changed in their visibility.
So here's how the turn order might go:
- Oracle gives directions (holding until both quasits move)
- Eagles are moved to that square, have a standard action to spend.
- Quasit is probably in Total Defense for the standard action, and then spend its move action. Unless the eagles are grappling it, of course.
A very messy version of things could happen since quasits are reasonably (15) INTelligent
- Oracle calls out location.
- Quasit hears call out and uses delayed action to move.
- Eagles blindly probe from last known location.
- Oracle waits until eagle initative, calls out.
- Eagles probe
The worst part is when they start trying to wait for each other and then initiative gets thrown out the window and we might switch to narrative mode to keep things a little more streamlined. Not only that, but this pretty much ignores that the Quasit has Detect Good (if either the summoner or oracle are good-aligned), and Cause Fear if a pair of hunting eagles come a-charging. It might even turn into a wolf to fight back and who knows how it all breaks down.
Doing more than 8bit tactics combat, we have to consider that a single quasit is most likely to just run away when this outnumbered, especially when the larger force hides itself. Now, depending on the situation, the quasit probably has some hidey-hole to exploit if it was somehow caught alone. If not, you sound the bugle and just watch the eagles hunt it down like releasing the hounds.
Let's tease the idea that the quasit attacks. Let's remember that this little demon has basic human smarts and can speak common so now we enter the quasit maybe trying to use stealth v. party perception. Now the Oracle needs a silent way or another language to tell where the quasit is (which it may get the gist of anyway and adjust). A silent way without the party seeing him if they aren't clustered. Then it becomes a stealth v. quasit perception if they don't want it to hide in the mists as well until finally this game of cheating Battleship gets a hit.
Ultimately there is one extremely limiting factor: Terrain. Sure the Oracle can see, but is there anything the quasit can hide behind not made of water? The eagles might be 60% faster, but they are also a few grades lower in maneuverability and a larger creature. One tightly winding tunnel and the quasit breaks free of the party.