According to the Hague Convention, as long as you have a pending case, they can't make a determination of custody and without that, can't assess child support. I realize this is regularly violated in Brazil, but technically, you should be able to freeze any cases/rulings at the local or state level until your Hague case is resolved. The area where it is unclear is whether this applies if the taking parent wins at the first round and the Hague case is at the appeal stage.
Article 16
After receiving notice of a wrongful removal or retention of a child in the sense of Article 3, the judicial or administrative authorities of the Contracting State to which the child has been removed or in which it has been retained shall not decide on the merits of rights of custody until it has been determined that the child is not to be returned under this Convention or unless an application under this Convention is not lodged within a reasonable time following receipt of the notice.
