When choosing compressor you should consider if it will be able to constantly provide desired pressure and level of air. Little airbrush compressors small compressors boast impressive pressure figures but actually poor flow rates enough limited to airbrush pistols. Actually if small compressor can push enough air it will likely overheat in couple of minutes.
Our laser cutter was included with little membrane pump (Smallest upon the picture), I really do not know very well what specifications it has but stream of air was extremely weak. It is enough if all you have to to do is engrave, although even then we got considerable smoke traces around engraved picture.
it had been rated for volume of ~2.4cfm and had a 1 gallon tank which can be filled to 58 PSI. It managed to give us continuous 1 bar (14 PSI) of pressure, but it had to run all the time and overheat badly within 5 minutes.
Third and Air Compressor For Laser Cutting current compressor we got was 6cfm rated belt driven device with 13 gallon container which is filled to ~100psi. With this unit we can run at constant ~2bar (31psi) without concern with overheating since compressor engine is not on constantly. Using reserve atmosphere in tank we can also force up to 3.5bar (50psi) for short jobs.
The best setup in my opinion would be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.