Times have moved on.
www.homebriefing.com is the one website flight plan filing facility which has confirmed they will handle FPs to anywhere, not just within Austria where they are based (Vienna ATC).
The addressing "should work" provided your FP contains at least one waypoint in each country. They then address it automatically using a database of addresses to
departure
destination
the FIS area(s) you pass through.
A great service, not free but cheap.
Rarely fails, but obviously it pays to telephone the departure airport anyway (before leaving home/hotel) to make sure they have the FP. I had it fail once, when flying to a little place in Spain and their system didn't send the FP to the correct big place nearby.
The above is for VFR flight plans. IFR flight plans in CAS use a different system; potentially much more involved and done via Eurocontrol. I will PM some details.
There is a bit of a quirk if you file a trivial IFR FP in the Uk and entirely OCAS; I don't think it reliably goes to Eurocontrol but could be wrong.