When choosing compressor you need to consider if it can constantly provide desired pressure and volume of air. Little airbrush compressors little compressors boast amazing pressure figures but actually poor flow rates adequate limited to airbrush pistols. Also if small compressor would be able to push enough air it’ll likely overheat in couple of minutes.
Our laser cutter came with small membrane pump (Smallest on the picture), I really do not know very well what specifications it has but stream of air was very weak. It is adequate if all you have to to accomplish is engrave, although even then we got considerable smoke traces around engraved picture.
it was rated for volume of ~2.4cfm and had a 1 Air Compressor For Laser Cutting gallon container which may be filled to 58 PSI. It managed to give us constant 1 bar (14 PSI) of pressure, nonetheless it had to operate on a regular basis and overheat badly within 5 minutes.
Third and current compressor we got was 6cfm rated belt driven device with 13 gallon container which is filled to ~100psi. With this unit we are able to run at constant ~2bar (31psi) without concern with overheating since compressor engine isn’t on continuously. Using reserve surroundings in tank we can also force up to 3.5bar (50psi) for short 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.