Presuming you could previously play back 4K footage without stutters, then yes, this does sound like dust in the gimbal. If it is you should be able to see the gimbal struggling / delaying as you input the command to rotate. Unfortunately, debris in the camera / gimbal assembly is a lot harder to get out and clean than it is to get in there in the first place, hence why we'd normally recommend using a landing pad for taking off in dusty environments....
I have no experience of doing this myself, and there isn't much reference online I can find, so will have to leave it to those who have done this before to tell you the best way to proceed.