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