Well, I've never played other PS games, so it may be in those too, but I do know it is at least in PSO2.
Anyway, we will call this an Enemy Burst, and it's based on the PSE Burst system in PSO2.
First off, the core of the system is to cause more enemies to spawn when certain conditions are met. These enemies will give boosted rewards, including increased EXP, Gold and item drops.
There would need to be some kind of meter to gauge your progress towards the Enemy Burst. For example, the meter may charge a small amount every time you hurt an enemy, and a larger amount when you kill them. You could get even higher charges by striking their weakness, inflicting states, or other things. You would lose a bit of the meter at the start of each turn, every time you take damage, have your weakness hit, are afflicted with a state, or, most especially, die.
Ultimately, this means that the faster you kill the enemies whilst exploiting their vulnerabilities, the higher chance of an Enemy Burst happening.
When you kill all enemies with the Enemy Burst meter full, you will trigger the Enemy Burst.
This will in turn spawn another troop without leaving the current battle, either specified by the troop you started with, based on random battle, or whatever. There would be a few ways to specify how you want your enemy burst to work. You would gain advantage and begin each turn for the remainder of the Enemy Burst with a huge agility boost, ensuring you always get the first action. The meter will continue to slowly drain, and the longer you keep it going, the more it drains per turn, but you can counteract this for a while by battling intelligently.
At the start of each Enemy Burst round, it would ask you if you want to continue or finish the burst, which will end combat with the rewards you have gained thusfar. This can be used to break out if you find yourself overwhelmed by your foes and close to death.
If you kill all of the enemies in the Enemy Burst round and you still have more Burst Meter remaining, you'll trigger a Cross Burst, which will spawn yet another troop in the same manner as before. You can then specify if you want the boosted EXP/Gold/Item drop rates to increase, and by how much, for each successive Cross Burst.
You could also use this method to create special enemies that you can only encounter if you get a certain number of Cross Bursts.
Eventually though, it will be impossible to keep the meter up, as the drain value increases per round, and the Burst will end, leaving you to finish off the remaining enemies...
...But then...
ONE MORE!
There would be a chance of triggering a One More, which would be determined by how many Cross Bursts you could keep up. This will give you a last boost to your Burst Meter, only a small portion, and one more troop to fight. It would also be possible to specify how much of a boost EXP/Gold/Items get during a One More, separate from the specified values for Enemy Burst and Cross Burst.
After the One More troop is defeated, whether you have meter left or not, the battle will finally end, and you will reap the rewards.
Also of note, is that it should be a function that is basically opt-in, where Enemy Bursts will not occur unless you're in a troop that is specified to be able to create Enemy Bursts.