All quarries that form Running Hill Pits are mapped as open access land with the exception of Quarry No 7 and 8. Access land gives a right of access on foot for quiet recreation, specifically including climbing as a permitted activity. The landowner is able to restrict access for up to 28 days at their discretion, but details of this must be notified on site and on the Natural England website (see 'CRoW Information' section below for more details).
Alternatively, a shoot may ask climbers to hold back temporarily whilst they move thorugh the area for safety reasons, after which you are able to access the area again. This approach is is only a slight inconvenience rather than a ban covering the whole day, so do what you can to work with the shoot.
Climbers have experienced problems with being asked to leave at the crag, presumably by the landowner or one of his agents, but no details were given. If there are no temporary restrictions in place, you have a legal right to access the quarries and surrounding access land for climbing and walking. If asked to leave, please ask for the person's contact details and pass them onto the BMC access team.