Updated : 25/04/2017
YES - Any interested party may ask for a judgment on parental responsibility made in one EU country to be recognised and enforced in another.
YES - In exceptional cases, if the child has a particular connection with this other country and if the transfer is in the child's best interests.
NO - The only court that can deal with matters related to your divorce/legal separation, including parental responsibility and visiting rights, is the first court where the request for divorce was filed.
So you will have to take your case to that court in Luxembourg – unless you can prove that the children have a particular connection to France.
YES - Because taking the children with you could make it difficult for him to take care of the children in line with the custody arrangement – in fact, legally, it might even count as child abduction.
You would also need his consent even if you had full custody, if only to respect his visiting rights.
However, another option could be for you to apply for a change in the custody or visiting arrangements – a court might grant a change if it thinks this would be in the children's best interest.