for British, from UK--BJ, you need one entry visa,

but from BJ--HK, British passport holder doesn't need entry  permit. For Chinese passport holder with PR, if the route is mainland China--HK--another country(not mainland China), Then you don't need a entry permit. Just need the flight tickets to prove your itinerary. But if your route is mainland China-HK--mainland China, then you certainly need an entry permit.