The conventional way to do this is Racking (Wikipedia) in which the wine is slowly syphoned from one vessel to another, leaving the sediment undisturbed.
This is better than trying to wash out the sediment with good wine.
In the case of wine, the sediment is called lees (Wikipedia again) , and it's mainly made up of dead yeast. In the early stages of making a batch of wine, there will be fruit residue in there too (I've just been making grape juice, and the fruit sediment looks a lot like lees).
The syphon has a spacer of some form at the bottom, to draw the wine off above the lees.
Most of my brewing stuff is in the loft, but one syphon was accessible, so I mocked it up with glasses taking the place of your demijohns or brewing bucket
Commercially, racking is also done by drawing off the wine just above the sediment, which is shown in two pictures in the Wikipedia article I linked (the first picture might show racking by syphoning, but it's not clear)
It is possible, at the final racking, to pass the wine through a filter paper in a dedicated holder. This is one method of clarifying wine which can also use additives to encourage suspended particles to settle out.