When choosing compressor you should consider if it will be able to 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’ll likely overheat in short while.
Our laser cutter was included with small membrane pump (Smallest on 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 significant 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 tank which can be filled to 58 PSI. It managed to give us constant 1 bar (14 PSI) of pressure, nonetheless 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 fear of overheating since compressor engine isn’t on at all times. Using reserve atmosphere in tank we can also drive 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.