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.
Open access land, designated under the Countryside & Rights of Way Act (2000) give area access rather than linear access as provided by public rights of way. It also gives a legal right of access specifically for climbing, as well as walking and other quiet recreation on foot.
Please bear in mind however that the landowner still has the right to restrict access for up to 28 days per year (often used on public safety grounds for shooting in moorland areas), and can also apply for longer term restrictions with Natural England (such as bans on dogs, or regular restrictions during particular times of year). It is important to check for these restrictions regularly as they can be added at short notice – all details for open access land in England can be found on Natural England’s website.