Do power amplifiers that use a dynamic power supply operate in the active region, i.e. as a voltage controlled current source? or do they operate in the saturation region? and how do the current and voltage waveforms look like?
Only switched amplifiers (classes D, E, F, etc.) work in "saturation region". This kind of amplifiers doesn't switch the power supply rails because they are intrinsically efficient.
The only amplification classes that switch power rails or use some kind of dynamic power supply are class G and H, which are basically class AB amplifiers (thus working in "active region") with an "enhanced" power supply whose purpose is to achieve higher efficiencies (although typically lower than class D) while keeping distortion low.
Class G/H is typically used in audio application by manufacturers like NAD, Emotiva, Sunfire, etc.