The embassy will send all documents to the Federal Office for Asylum and Foreign Services (Bundesamt für Fremdenwesen und Asyl -BFA) in Austria. The documents will be submitted to a judge, it is often the same judge that confirmed your notification, he is familiar with your story.
It is possible that the embassy or the Federal Office (BFA) will interview your family. The BFA has to be sure that you really are family. You can show the BFA pictures or other documents. It is possible that the BFA will require a DNA test.
The BFA will decide and send a notification to the Embassy, which will inform your family. Is the final decision positive? Your family will get a visa. The family has 4 months to reach Austria. Is the decision negative? The embassy will inform your family.