When choosing compressor you have to consider if it can constantly provide desired pressure and level of air. Little airbrush compressors small compressors boast amazing pressure figures but actually poor flow rates enough only for airbrush pistols. Even 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 upon 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 had been rated for volume 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 run all the time 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 continuous ~2bar (31psi) without concern with Air Compressor For Laser Cutting overheating since compressor engine isn’t on all the time. Using reserve air 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.