When choosing compressor you have to consider if it can constantly provide desired pressure and level of air. Small airbrush compressors little compressors boast amazing pressure figures but actually poor flow rates sufficient only for airbrush pistols. Actually if small compressor would be able to push Air Compressor For Laser Cutting china enough air it’ll likely overheat in short while.
Our laser cutter was included with little membrane pump (Smallest on the picture), I do not know very well what specifications it has but stream of air was very weak. It 
is adequate if all you want to do is engrave, although actually then we got significant 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 operate all the time 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 constantly. Using reserve atmosphere in tank we are able to also press up to 3.5bar (50psi) for brief jobs.
The best setup in my opinion would be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.