Filemaker Pro allegedly can talk to MySQL through MyODBC, but the standard implementation supports only IMPORTING into Filemaker, not exporting. This is a Filemaker shortcoming, not a MySQL shortcoming. There may be some third-party Filemaker tools that can help you.
We use Filemaker Pro at work to manage data (news stories), and are pushing the results to MySQL by exporting it as XML and then importing it through PHP. That works fine for pushing structured objects, but not so fine for synchronizing a complex relational data model.