Found this post on another site:
Cleaners like Windex contain ammonia and will erode the chrome plating on an airbrush. They work well, but you pay a price in the long run. I use plain water in a spray bottle to clean my gravity feed brushes between colors. Shoot it in the bowl, spray it through the brush into a trash can, then backwash the brush by holding your finger over the tip and pulling the trigger. Gets it clean in less than a minute. At session's end, I disassemble the brush and clean everything with solvent. Every few months, I take the brush apart and soak it overnight in airbrush cleaning solution. If you prefer to mix your own or use an alternative cleaners, check out airbrush.com's Tips and Tricks section. There are dozens of home brew cleaners/reducers/thinners suggested there.
Pretty good advice but....beware
... many cleaners will degrade the chrome finish inside the paint bowl and when the chrome is gone paint sticks to the bare metal and restricts paint flow thru the airbrush. Best to stick with water or the manufacturers recommended cleaners.
I use water and a round paintbrush about the size of your finger...you can find them at WallyWorld. Rinse the bowl with water once...pour the dirty water out...refill the bowl with clean water and use the paintbrush to clean the bowl thoroughly...pour the dirty water out...fill the bowl with clean water to about 1/3 of the bowl and spray into your cleaning pot...do that twice. Add a drop of clean water into the bowl and spray onto a white towel or sheet of paper...if it sprays clear you're done. Sometimes you may need to use the paintbrush to carefully clean the tip of the airbrush. For a monthly cleaning use the manufacturers recommendations for maintenance. Don't fudge on the maintenance and your airbrush will perform much better.