Flaperon washes up on Reunion Island

[peekay4]

But what part of that gave them the "responsibility or right" to conduct the investigation?

The basic Annex 13 rules:

1. If an accident occurred within the territory of an ICAO signatory, then that country is responsible for the investigation ("State of Occurrence")
2. If an accident occurred outside the territory of any country -- or if the location of the accident cannot be definitely established -- then the country where the aircraft is registered becomes responsible for the investigation ("State of Registration").

(There are exceptions to the above rules, involving accidents in non-ICAO countries, etc.)

AF447:

• Since the accident occurred in international waters, due to #2 France as the "State of Registration" was responsible for the investigation.

MH370:

• If the location of the accident were determined to be in Australia, then due to #1 Australia would be responsible for the investigation
• Alternatively, if the location of the accident were determined to be in international waters -- or if the location of the accident cannot be established -- then due to #2 Malaysia would be responsible for the investigation.

Note: Any country may delegate an aircraft investigation to another country (by mutual agreement), in whole or in part.