When choosing compressor you should consider if it can constantly provide desired pressure and volume of air. Small airbrush compressors small compressors boast impressive pressure figures but really poor flow rates sufficient limited to airbrush pistols. Even if small compressor can push enough air it’ll likely overheat in short while.
Our laser cutter was included with little membrane pump (Smallest on the picture), I really do not know very well what Air Compressor For Laser Cutting specifications it has but blast of air was very weak. It is enough if all you have to to accomplish is engrave, although even then we got considerable smoke traces around engraved picture.
it had been rated for level of ~2.4cfm and had a 1 gallon container which can be filled to 58 PSI. It managed to give us constant 1 bar (14 PSI) of pressure, but it had to operate all the time and overheat badly within 5 minutes.
Third and current compressor we got was 6cfm rated belt driven unit with 13 gallon container which is filled to ~100psi. With this unit we can run at constant ~2bar (31psi) without fear of overheating since compressor engine is 
not on constantly. Using reserve air flow in tank we are able to also push up to 3.5bar (50psi) for brief jobs.
The best setup for me would be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.