Wiktionary
vb. 1 Of a liquid; to move slowly down to lower substrate levels. 2 (context idiomatic English)(by extension) Of information, or resources; to move slowly down to lower levels of an organisation, or population.
Usage examples of "filter down".
The faint trickle of liquid over rocks was reassuring in the way that the sound of water could be, and after an hour he came upon a break in the tree canopy that allowed sunlight to filter down on the stream in slanted shafts.
She swung her dust muffler around her shoulders, keeping the hood and scarf-filter down so her face remained visible, then tossed Sulu’.
Alert to rumors, which were always helpful, he waited for news to filter down through the grapevine.
I left my pack propped against a big rock at the trailside and went with our water bottles and the filter down to the pond edge and filled up.
It took the better part of a day for ten gallons to filter down through the system to the collection tank.
But it'll take a while for the word to filter down to the rank and file.
Don't expect plants, there's not enough light that far from the sun, even if it could filter down through the layers of ice.
On as small a ship as this, the upset might filter down through the ranks, even to his lowly station.
Bits of ash and debris were beginning to filter down around them in a fine rain.