G'day Gums,
The venerable Douglas/Boeing 717, a DC9 on steroids, works thus:
Selecting Autoflight on the takeoff roll once symmetrical power increase is observed engages the Auto-throttles to the target takeoff power, be it REF power or flex/derate.
Passing the acceleration altitude entered into the FMS as part of the pre-departure procedures will see the take-off power reduce/increase to climb power, with the aircraft accelerating towards (usually) 250 kias. During this phase the AFS uses pitch to accelerate to or maintain 250 kias, and once through transition altitude to further increase to enroute climb speed. The Flight Mode Annunciator panel (FMA) will indicate SPEED on PITCH & T/O or CLIMB THRUST.
Once cruise level is reached the FMA will change to indicate SPEED (FMS or manually selected speed) on THRUST and LEVEL HOLD, and the auto throttle system will adjust power within the bounds of its capabilities to maintain the set speed.
When commencing a normal descent, the auto throttle system will for the most part continue to use thrust to manage speed, whereas pitch will manage the flight path. Should the aeroplane get higher than the computed flight path, the auto throttle will reduce power to idle, and may increase the IAS/MN to the maximum allowable minus a margin that is too small for comfort. On regaining profile the auto throttle will revert to SPEED on THRUST
A normally flown instrument approach or visual approach will see the SPEED on THRUST relationship remain, and in the flare the thrust will be reduced to idle, with reverse thrust being manually selected.
In any of the above modes, if the FMS detects the speed at Vmin - 5 kts, the auto throttle will advance the thrust, or if it had been turned off (but not broken) it will engage the auto throttle, and use a combination of thrust and pitch to maintain Vmin-5 and sacrifice altitude until the pilot intervenes and selects a faster speed.
In essence the 717 philosophy is to engage auto throttles during takeoff and leave them engaged until landing.
I think in my 8 years on type I've only had two occasions where the auto throttle system didn't work and both were a result of FADEC failures.