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.