As far as I am aware, the ONLY joint Shooting team in the CF is the MARLANT Combat Shooting Team. If you look at the photo of the QM winner being chaired off last year, you'll see a Sailor in the Chair, and the front four carrying the chair are: Navy (Reg), Navy (Res), Army (Reg), Airforce (Reg), followed by an Army Reserve, and members of the British Army Combat Shooting Team.
The MCST has been supporting and promoting this spirit since I got involved with the team in 1993.
The Bisley team (CF Combat Shooting Team) is composed of a selection of shooters from the top 30 at the previous year's national matches, which is not necessarily going to be tri-service.
Every other unit team is unit specific.
NS