When choosing compressor you need to consider if it can constantly provide desired pressure and volume of air. Little airbrush compressors little compressors boast amazing pressure figures but really poor flow rates adequate only for airbrush pistols. Actually if small compressor can push enough air it’ll likely overheat in short while.
Our laser cutter came with small membrane pump (Smallest on the picture), I do not know what specifications it has but blast of air was extremely weak. It is sufficient if all you have to to accomplish is engrave, although also then we got substantial smoke traces around engraved picture.
it was rated for volume of ~2.4cfm and had a 1 gallon tank which may be filled to 58 PSI. It were able to give us continuous 1 bar (14 PSI) of pressure, nonetheless it had to run all the time and overheat badly within five minutes.
Third and Air Compressor For Laser Cutting current compressor we got was 6cfm rated belt driven unit with 13 gallon tank 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 continuously. Using reserve surroundings in tank we can also press up to 3.5bar (50psi) for brief jobs.
The best setup in my opinion will be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.