Rear gutters collect a lot of water
Where the roof runs into the back of a wide chimney, the lead detail has to carry water around the stack. Splits, blocked channels and poor laps can all send water under the covering.
The stack itself may be taking in rain
Open mortar joints and cracked flaunching can soak the masonry. Water can then emerge lower down, sometimes making sound leadwork look guilty.
Unused flues can stay damp in dry weather
If moisture appears around a chimney breast after cold nights but not after rain, condensation inside an unventilated flue is worth considering.