Pulsing outputs on power off are an annoying fact of life and depends on the specific equipment; it can also be seen with status LEDs flashing on as power is turned off in some cases.
What happens is power turns off but the supply rails do not drop instantly nor always at the same time throughout the entire circuit. For the pulsed LED case; if the cathode voltage falls to 0V as power goes off, but the +V the anode connects to has some juice left in it, the LED will light and then dim as +V also drops to 0V. Other circuits can behave in a similar way, outputs going high as their inputs go low, then slowly dropping as power actually disappears.
Similar glitches can happen at power on, especially if something behaves differently to how it does when fully initialised which may be some time later.