What do people recommend for cleaning the bore of PCP and Break Barrel rifles?

They don't require much cleaning....unless it's new. Then the barrel on most will be filthy. But air rifles just don't require very much in the way of regular barrel cleaning really. I normally use a Dewey rod with patches soaked in Ballistol. Wet patch, then dry, wet patch, then dry.....etc. This until patches come out clean. Ballistol cleans plenty well, lubricates plenty well, and is safe for o-rings.

There is no undisputed right way.....everyone has their preference. But, you will find common/similar techniques amongst experienced air gunners that you should pay attention to. Because there are absolutely some DO NOTS.
 
Ballistol for everything. Use it with a Crown Saver pull through for cleaning, and it's a great slug/pellet lube too.

Crown Saver or Patchworm, using a rod inside the barrel always makes me nervous. Germany invented Ballistol in the 20th century and it is still being produced today. Works on everything, metal, wood, leather rubber or plastic. Cleans the hell out of my rifle and shotgun barrels.