You can use oil paints through your airbrush here are some articles from our archives oil paints discussion
also remember To avoid problems when switching from one paint system to another (ie. from oil base to waterbase), spray a little (rubbing) alcohol through your airbrush when you are finished cleaning.
Also always take out the needle at the end of your cleaning procedure, wipe it down, reinsert, take it out again. If there is paint on the needle, the airbrush is not as clean as you thought.
another group of thoughts thoughts one thoughts two
Here is a WIP done by an amazing artist by the name of Paul Corfield