Worth noting that some French show caves actually have airlock double doors, both at the entrance and separating different areas of the cave, specifically to prevent draughts and restrict the airflow. This seems to be on conservation grounds, i.e. to prevent changes in the cave microclimate which might affect either specific formations or maybe cave life. I came across this in the Vercors and was quite surprised, remembering radon issues in the UK.
Don't know if the same situation applies in other French show caves or in show caves in other countries.