At what point does the throttle control RPM on a constant speed prop?
What happens is when you're on final and you move the prop to MAX RPM, the prop governor moves the blades finer until the RPM is at red line, then governs RPM there by adjusting blade angle as required.  As you reduce throttle, the governor moves the blades finer and finer trying to keep the RPM that has been set (maximum or redline RPM in this case). 

As throttle is further reduced the blades hit the fine pitch limits of the propeller hub and can't go any finer, so the governor can't unload the engine any more to keep the RPM up.  As you continue to reduce throttle you basically now have a fixed pitch prop with very finely pitched blades and down comes the RPM.  As long as throttle is below the level that will speed the prop up above redline (or whatever the governor setting is), RPM will vary directly with throttle, like a fixed pitch prop engine.

Add power and RPM rises until it hits the governor setting, then the governor starts to coarsen pitch to load up the engine, to keep the RPM at the setting.

The point at which the prop "comes off the governor" is determined by the pitch limits of the blades within the hub.  The part the mechanic can adjust is the RPM set by the governor when the prop lever is at MAX, which is normally redline RPM +/-.