Breifly, the next-door neighbour has plastic, square guttering on their side
of the two semi's. We have the original round guttering which feeds a cast iron
outlet into the downpipe. The gutter leaks badly where the square and round
types meet.
It turns out that while the new guttering does actually have a square-round
adapter, this has merely been placed into the downpipe's outlet with no sealant
or physical attachment - it just flaps around in the wind and rain.

While up t' laddter today I decided to see what remedies were possible. The
round outlet has two screw-holes in its base. The side that goes to the half-
round guttering is screwed onto the outlet, the side that goes to the adapter
isn't - it's just loose. This appears to be because of a mismatch in the
"systems": modern outlets have a rubber seal to make a watertight joint, the
original cast iron outlet expects the guttering to be screw in and doesn't
have a gasket.
The plan is to bed the loose end of the adpater into some adhesive goop and
possibly to drill a hole up through the adapter to form a physical attachment
(as opposed to simply an emotional one). Though I'm a little dubious whether
a hole + metal screw would stand the test of time.

So, the question is: do I need to consider thermal expansion in the joint?
Would an adhesive ultimately fail due to movement, and/or should a screw-hole
in the bottom of the adapter be elongated to allow expansion along it's length?