Normally, it is recommended to clean out your airbrush with the same thinners you originally mixed your paint with. However, sometimes dried paint can be a little stubborn to remove, especially in hard to reach areas inside your airbrush. 

Brands like Tamiya, Gunze and AK Interactive have specially formulated thinners to remove all the gunk. 
Hobby Supplies Paint Thinners


Although these airbrush cleaners are made for general purpose cleaning, use with caution and make sure you have sufficient ventilation as they can be harmful to breathe in. All thinners and cleaners will also cause rubber seals to deteriorate over time, so with frequent use they may need to be replaced.
Airbrush Cleaners