I use the dyed cheesecloth in number of places; to mute colour that is too strong or to add a little texture in an otherwise dull spot. Dying it couldn't be easier.
After stuffing whole pieces into a small jar, I add the rinse water from the empty bottles of paint. They can be mixed on the spot.
Only a little liquid is necessary. Cap the jar and agitate the contents until all the white is gone.
The more it is agitated the more homogenous the colour will be.