Both our old Jazz I'm scrapping and the replacement have this. Even though we live in a most extreme area, by the sea, with constant salt spray, I suspect it is a common problem ?
Incidentally, both cars are 2006, the scrapper owned from new and just failed its MoT on a rusty sill (noisy gearbox and slipping clutch too ....). The replacement, for 1200 pounds is in comparison immaculate. Both cars in Cornwall for most of their lives.
I've not tried removing the trim, but if you do I'm sure you can improve things at this stage. I suspect there is a sharp edge with little paint on it underneath the rubber ?
The picture I've attached shows a hole through the rear door where it is almost a single skin. It was so bad (I thought it was the black trim bubbling at first), but it was terminal rust. A place to watch !