The State in which you infringe will be the one who brings the action. In the case of the French DGAC they normally report the infringement to the UK CAA if the infringer is a G registered aircraft, or if the pilot has a UK licence and is flying a F registered aircraft!
The UK CAA invite the individual for a voluntary interview; the individual has the right to refuse, to go alone or with a representitive. The CAA has no power to do anything but may offer the infringer some sort of remedial package such as additional training and or a test. If the infringer accepts this, it will be reported to the DGAC and the case will most likely be closed.
If the Infringer opts to ignore the CAA then the DGAC will invariably take action in the French courts; fines can be up to 15000 Euros for a TRA infringement or more if deliberate.