If you google PIP advice the numbers that come first are all commercial advice lines that charge a fee to give advice,do not use them!
The official government PIP claim / advice line is on this link
https://www.gov.uk/pip/how-to-claim
But reading through the claim information it really looks as though you can not continue to claim it unless you are a full time UK resident.
But just phone the claim line they will tell you straight away if you can claim from France.
But I still think that once you move you will need to claim French benefits.