My simplified view:
Crossing the threshold you want to be at a certain AIRSPEED. This equates to a certain GROUND speed.
GS Mini makes sure the aircraft maintains at least this speed across the ground during the approach.
Example.
Airspeed = 130 kts. Headwind 5 kts. Ground speed = 125 kts.
Say at 500 ft. Headwind = 40 kts. Airspeed would 165kts but Groundspeed still = 125 kts.
Therefore when the headwind suddenly drops off, the aircraft already has the correct groundspeed which combined with the tower headwind is the correct amount of energy.
FIS.