When choosing compressor you need to consider if it can constantly provide desired Air Compressor For Laser Cutting pressure and level of air. Little airbrush compressors small compressors boast impressive pressure figures but really poor flow rates enough only for airbrush pistols. Actually if small compressor would be able to push enough air it’ll likely overheat in short while.
Our laser cutter was included with small membrane pump (Smallest on the picture), I really do not know very well what specifications it has but blast of air was very weak. It is adequate if all you have to to do is engrave, although also then we 
got considerable smoke traces around engraved picture.
it was rated for level of ~2.4cfm and had a 1 gallon tank which can be filled to 58 PSI. It managed to give us constant 1 bar (14 PSI) of pressure, nonetheless it had to operate on a regular basis and overheat badly within five minutes.
Third and 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 fear of overheating since compressor engine is not on constantly. Using reserve surroundings 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.