There is more to it than simply who takes lead of the investigation and produces the report. There were (4) French nationals on board, and the interim report quite unambiguously stated that provided the conclusions derived from the Inmarsat data are correct, only intentional human intervention could have placed the aircraft on its final course.
Technically that conclusion should call for a homicide investigation, and as the accident happened in international waters, any claimant party could launch it. However in most jurisdictions some physical evidence is required that in fact the homicide did happen (disappearance of a person is not enough). The French have just acquired that crucial piece of evidence (I think there is no doubt this is in fact a piece of MH370), so there will be a lot of wrangling and arm twisting in the background on who has custody and under what terms can other parties gain access.
As usual, the most needy and deprived will gain the windfalls - the lawyers...