Applying my limited understanding of the subject:
If a source of light falls into a black hole, it appears to have its time dilated in such a way so as to never reach the event horizon. But it also means that any two events happening on the source become infinitely stretched in time, as seen by an external observer.
If the two events are two consecutive emissions of a photon, then as it asymptotically approaches the event horizon, the time between the emissions becomes inifitely long. As a result the source appears to grow dimmer(flux goes down) across all wavelengths, even before taking into account the gravitational redshift.