The slaved gyrocompass does the work, but slowly. The plus/dot window shows the electronic flux valve error to the compass - the signal the slave mechanism sees as the correction to the drifting gyro. In other words, the little window is the electronic indicator of the "truth" as the flux valve sees it. The flux valve is an electronic mag compass. To help the flux valve, many gyrocompasses have a latitude set on the case, so the mechanic can give it a guess as to the dip angle and variation when it is installed. This helps it adjust faster.
In normal operation, a swinging indication of plus-dot tells you that the compass is aligned with the flux valve. Since the change in mag variation is slight when you fly around from place to place, the flux valve is very weak in its ability to update the gyrocompass. This is done on purpose, so that flux valve failures don't cause you to fly to London when you meant to go to Paris. I don't know the rates, but they are down in the 1 degree per minute neighborhood.
The manual slew rate is faster, maybe 15 degrees per minute, so you can command a correction and the manufacturer can count on you to make a good judgement, so he is not blamed for that error in a Failure Hazard accounting of his system.
An unslaved gyro drifts at a steady rate as the earth turns, since the gyro is rigid in space and the earth is moving it and rotating it. That error is 15 degrees per hour. Most gyrocompasses have friction in their mechanisms, so they drift much faster than that.
Lindbergh flew the atlantic with his most modern piece of equipment as the Earth Inductor Compass, basically a flux valve indicator. The fact that he was 1/4 mile off course when he hit the Irish coast shows how good we can be if we know our equipment. Of course, if you are as good as Lindbergh, you're set already!!