If your spouse or partner — your children’s other parent — works in your home country, you are entitled to child benefits there. If the amount of corresponding benefits in the country where you work is higher, the country where you work should pay a supplement corresponding to the difference between both benefits.
If your spouse or partner — your children’s other parent — works in the same country as you or doesn't work at all, you are entitled to child benefits in the country where you work.
If you and your partner work in different countries but none of you works in the country where your children live, you'll receive family benefits from the country where the highest benefits are paid.
NO — you cannot receive family benefits twice for the same period and for the same child. There are "priority rules" in cases of overlapping entitlements.
If your double entitlement results from the fact that you work in one country and live in another, you will only receive benefits from the country where you work.
If you work in both countries, the country where you are insured will pay you family benefits.
YES — if your ex-husband doesn't use the family benefits he receives to maintain your children, the authority providing the benefits can decide to pay them directly to you, as the person who is actually maintaining the family.
If you want them to do so, contact the family benefits authority of the country where you and your children live. They will then contact the authorities responsible for paying the benefits in the other country.
In that case, the benefits will be paid by the country (France or Luxembourg) where the highest benefits are paid.