When choosing compressor you have to consider if it can constantly provide desired pressure and volume of air. Little airbrush compressors little compressors boast impressive pressure figures but actually poor flow rates enough only for airbrush pistols. Even if small compressor can push enough air it will likely overheat in short while.
Our laser cutter was included with small membrane pump (Smallest on the picture), I do not know what specifications it has but stream of air was extremely weak. It is adequate if all you have to to do is engrave, although also then we got substantial smoke traces around engraved Air Compressor For Laser Cutting china picture.
it had been rated for volume 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 run constantly and overheat badly within five minutes.
Third and current compressor we got was 6cfm rated belt driven unit with 13 gallon tank 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 constantly. Using reserve air flow in tank we can also push up to 3.5bar (50psi) for brief jobs.
The best setup for me will be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.