Stain on joining wall just below porch roof

Water appears to be getting into our house causing a stain on the joining wall of our semi detached house.

The wall is never wet or damp to the touch, but the stain has been gradually working it's way down the wall becoming more noticeable.

Inside - you'll see the stain on the joining property wall. The plaster is minimally raised in parts to the touch, but doesn't appear to be seperating from the wall at all.

Exterior - there is a gap on both the porch roof and a vertical gap joining with our neighbours house. It's unclear what purpose these gaps serve, e.g. for air flow. Could this allow a channel of water to go behind the exterior render, especially if it pools on the flat roof?

What could be the most likely cause and options be for repairing?