There is no answer to the general question because there are use cases where the answer is yes and use cases where the answer is no.
For example, there seems to be agreement in this thread that controls on a steering wheel for a car is not ideal.
However one can argue that controls on the joystick of an F16 are not only a good idea but one could not safely maneuver the aircraft and engage weapon systems at the same time without it:
As you can see in this diagram the joystick (labeled "sidestick controller") has a primary function of controlling the aircraft's attitude and secondary controls for setting trim, weapons systems, display systems, autopilot, and defensive countermeasures.
On the other side of the cockpit the throttle has a primary function of controlling the engine's thrust with secondary functions for radio communications, radar, etc.
In this configuration both hands are engaged with primary flight functions while individual fingers are manipulating controls on these moving controls to engage secondary flight functions or primary weapons functions.
In civilian aviation you have controls on moving controls. The primary jump ship at my drop zone (a Cessna 208B Grand Caravan) has a yoke with a radio button on the upper left side which allows the pilot to engage the radio to communicate with air traffic control without taking his left hand off the yoke leaving his right hand free to adjust trim or throttle (both primary actions necessary for maintaining flight).
With many broad design questions the answer is often it depends.
However in the case of aircraft, neither a joystick or a yoke have the same limitations as a steering wheel (their range of motion never puts the user's hand in a position where they can not easily engage the secondary controls). So the primary arguments against controls on controls for automobiles do not apply to aircraft. As a result, controls on moving controls are a good idea for aircraft.