I don’t think “staking a claim” comes into it, does it? By default the estate goes to the “reserved heirs”. If there are no reserved heirs then it is offered to more distant blood relatives. If there are no relatives at all then it goes to the State.
It’s the notaire’s job to track down and contact the relatives.
Non relatives have no claim, even a long-term cohabiting partner cannot inherit if there is no official relationship ie marriage or pacs, and no will naming them as heir.
The relatives will be required to prove their relationship via birth certificates, ID etc.
As far as the French property is concerned I don’t see it would make any difference. In the absence of any will, the French property would be dealt with by the notaire in accordance with French succession law, regardless of whether the proprietor was resident in France or not.