When choosing compressor you should consider if it will be able to constantly provide desired pressure and volume of air. Air Compressor For Laser Cutting little airbrush compressors small 
compressors boast impressive pressure figures but really poor flow rates sufficient limited to airbrush pistols. Also if small compressor would be able to 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 what specifications it has but stream of air was very weak. It is sufficient if all you have to to do is engrave, although even then we got significant smoke traces around engraved picture.
it was rated for level of ~2.4cfm and had a 1 gallon container which may be filled to 58 PSI. It were able to give us constant 1 bar (14 PSI) of pressure, but it had to operate constantly and overheat badly within five minutes.
Third and current compressor we got was 6cfm rated belt driven unit with 13 gallon container which is filled to ~100psi. With this unit we can run at continuous ~2bar (31psi) without fear of overheating since compressor engine isn’t on continuously. Using reserve air in tank we are able to also drive 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.