I don't completely understand the "brief period of time" in your question. Is this period followed by maintenance to regain response? is it during a run of multiple samples whereby you have a normal response in the begging then the MS becomes unresponsive for a few injections then it goes back to normal before the last few injections?
This is important to diagnose the problem.
If it is the first scenario, then it is clear that there is a problem that whatever maintenance that you are doing is fixing it.
If it is the latter, there could be many reasons. Do you have enough sample in the vial that the autosampler needle can withdraw from? Are you experiencing any bubbles in the lines during the run? Did you completely flush the system with fresh mobile phases for enough time to push out old stuff from the lines? I would also check the capillary tubing in the MS sprayer and transfer line to make sure the flow isn't obstructed by salt deposits. Make sure to clean the area around the sample cone in the MS, wipe the surfaces by solvents that contain 5% or more of formic acid.