How do you use an airbrush?

The simplest and quickest way to do a basic cleaning is to spray the solvent for the medium you are using through the brush. So for any water based paint or dye you simply put the stem in water and spray until it runs clear. If you do this in a sink you can keep spraying until you are sure any residue of paint has been blown through. For media that uses other solvents, you can do the same, though generally it's not cost efficient to blow a lot of solvent this way. But it works as a temporary cleaning job.

In all cases, to do it right you need to disassemble the brush and clean the parts in solvent.