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 sufficient limited to airbrush pistols. Actually if small compressor can push enough air it’ll likely overheat in short while.
Our laser cutter came with little membrane pump (Smallest on the picture), I really do not know very well what specifications it has but blast of air was extremely weak. It is sufficient if all you want to do is engrave, although even then we got considerable smoke traces around engraved picture.
it was rated for level 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, but it had to run at all times 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 are able to run at continuous ~2bar (31psi) without fear of overheating since compressor engine is not on on a regular basis. Using reserve air in tank we can also press up to 3.5bar (50psi) for brief jobs.
The best setup in my Air Compressor For Laser Cutting china opinion 
would be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.
