Braking is as you surmise: linear recoil assist
Compensation is muzzle deflection control.
It can help keep a muzzle down or from moving to the side and angles in between depending on type of comp used, the indexing of the comp, and general ability of the shooter to hold the firearm still.
EG: regardless of vertical rise/flip, right handed rifle shooters will always have some level of rightward muzzle motion as all the support is at the left, rear and bottom. Left handers end up with the muzzle moving left.
The muzzle simply takes the path(s) of least resistance.
The better the compensation, the less movement.
Shooter still has to actually have the ability to keep things fairly stable to begin with though, as a muzzle device won't be able to compensate for technique errors. They just enhance already employed good skills.
As in someone who is barely capable of holding the rifle steady even before firing or who gets rocked back on thier heels after is not going to suddenly be capable of 1/4 second split controlled doubles.
They are an assisting tool, not a magical device
