When choosing compressor you need to consider if it can constantly provide desired pressure and level of air. Small airbrush compressors little compressors boast impressive pressure Air Compressor For Laser Cutting china figures but actually poor flow rates enough only for airbrush pistols. Actually if small compressor would be able to push enough air it will likely overheat in short while.
Our laser cutter came with small membrane pump (Smallest on the picture), I really do not know what specifications it has but stream of air was very weak. It is adequate if all you have to to do is engrave, although also then we got substantial smoke traces around engraved picture.
it had been rated for level of ~2.4cfm and had a 1 gallon tank which can be filled to 58 PSI. It were able to give us constant 1 bar (14 PSI) of pressure, but it had to operate on a regular basis 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 isn’t 
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.