When choosing compressor you need to consider if it will be able to constantly provide desired pressure and level of air. Small airbrush compressors small compressors boast impressive pressure figures but actually poor flow rates adequate only for airbrush pistols. Also 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 upon the picture), I do not know what specifications it has but blast of air was extremely weak. It is sufficient if all you have to to do is engrave, although even then we got considerable smoke traces around engraved picture.
it was rated for volume of ~2.4cfm and had a 1 gallon container 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 at all times 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 are able to run at constant ~2bar (31psi) Air Compressor For Laser Cutting china without concern with overheating since compressor engine is not on continuously. Using reserve surroundings in tank we can also drive up to 3.5bar (50psi) for brief jobs.
The best setup in my opinion will be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.