I doubt you need me to tell you that. They are involved by default when an accident or incident occurs to a French-registered aircraft, over French territory or if the aircraft was built in France. In the first and third cases, they are usually guests of the host investigating agency if the accident happened overseas. Sometimes they are called in by other countries' agencies as an independent assessor.
In the case of the previous UAS incidents involving Airbus widebodies, their remit only extended as far as the host nations would allow - and none of them demanded action over and above that which was being done prior to AF447.