As a cross-border worker you are entitled to rights in your country of residence as well as in your country of employment (see priority rules). You must therefore apply in both countries. If an entitlement to family benefits exists in 2 countries, the differential supplement is the difference between the benefits paid in country A and the higher benefits in country B.
To obtain it, simply fill in a child benefits application form and attach the amounts paid by the fund in your country of residence and the other required documents.