When choosing compressor you need to consider if it can constantly provide desired pressure and level of air. Small airbrush compressors little compressors boast impressive pressure figures but really poor flow rates Air Compressor For Laser Cutting sufficient limited to airbrush pistols. Even if small compressor can push enough air it’ll likely overheat in couple of minutes.
Our laser cutter came with little membrane pump (Smallest on the picture), I do not know what specifications it has but blast of air was extremely weak. It is adequate if all you have to to do 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 gallon tank which can be filled to 58 PSI. It managed to give us constant 1 bar (14 PSI) 
of pressure, nonetheless 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 continuous ~2bar (31psi) without concern with overheating since compressor engine isn’t on all the time. Using reserve air flow in tank we are able to also force up to 3.5bar (50psi) for brief jobs.
The best setup in my opinion would be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.