Yes, yes. I know. Flat roofs are the spawn of Satan, and the nearest
thing to a solution is to rip the whole thing off and start again.
However, in the real world, the problem isn't really bad enough to
justify spending that much money.

I have a fairly modern (~2000) garage built back into the hillside.
The solid concrete roof slopes back towards a corner where it drains
into a downpipe that ends up in the public drains. There is a lot
of water coming through the roof around the outside of the downpipe.

The concrete is covered in a white(ish) material like rubber, plastic,
or lino, and this is then covered in something like carpet underlay,
with stone chipping on top.

What would the panel recommend as a suitable gunge to apply (either
with a brush, or a sealant gun) over the joints in an attempt to seal
them? Also, should I use expanding foam to fill the gap between the
downpipe and the concrete (from below) ?

(Final complication: The garage is in Germany, so the construction
and/or materials may well be unusual for the UK).