When choosing compressor you need to consider if it will be able to constantly provide desired pressure and level of air. Little airbrush compressors small compressors boast amazing pressure Air Compressor For Laser Cutting figures but actually poor flow rates sufficient limited to airbrush pistols. Even if small compressor can push enough air it will likely overheat in couple of minutes.
Our laser cutter came with small membrane pump (Smallest upon the picture), I do not know what specifications it has but blast of air was very weak. It is adequate if all you have to to accomplish is engrave, although also then we got substantial smoke traces around engraved picture.
it was rated for level of ~2.4cfm and had a 1 gallon container which may be filled to 58 PSI. It managed to give us continuous 1 bar (14 PSI) of pressure, but it had to operate at all times 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 fear of overheating since compressor engine is not on at all times. Using reserve air flow 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, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.