When choosing compressor you should consider if it will be able to constantly provide desired pressure and level of air. Little airbrush compressors little compressors boast amazing pressure figures but actually poor flow rates adequate limited to airbrush Air Compressor For Laser Cutting china pistols. Even if small compressor would be able to push enough air it’ll likely overheat in short while.
Our laser cutter was included with small membrane pump (Smallest on the picture), I do not know what specifications it has but stream of air was very weak. It is enough if all you want to accomplish is engrave, although also then we got significant smoke traces around engraved picture.
it was rated for volume of ~2.4cfm and had a 1 gallon tank which can be filled to 58 PSI. It were able to give us continuous 1 bar 
(14 PSI) of pressure, nonetheless it had to operate all the time and overheat badly within 5 minutes.
Third and current compressor we got was 6cfm rated belt driven device with 13 gallon tank which is filled to ~100psi. With this unit we can run at continuous ~2bar (31psi) without concern with overheating since compressor engine is not on at all times. Using reserve air in tank we are able to also drive up to 3.5bar (50psi) for brief 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.