For enamel or lacquer paints I blast some white spirit through to get rid of any excess paint in the airbrush.

Followed on with Cellulose Thinners until it sprays clear into a white tissue.

Then to finish I blow through some dedicated air brush cleaner to clean and protect the seals and 'O' rings

Acrylics    Water or Acrylic Thinners       Cellulose Thinners        Dedicated Airbrush cleaner

I use a brush loaded with thinners to clean the nozzle area and a wooden cocktail stick used very carefully for any stubborn bits in that area.