What is the process of appointing a new Governor-General if the current Governor-General fails to uphold his duties?
Interesting question, thank you Chloe-Jane. The Governor-General is appointed by the King on the advice of the Prime Minister. The Governor-General’s appointment is at ‘the King’s pleasure’, that is, without a fixed term. In practice, most appointments are for around 5 years. Sometimes this term is extended.
A Governor-General can be dismissed by the King before their term is complete. By convention – tradition – this may only be on the advice of the Prime Minister. No Governor-General has ever been dismissed.
If a Governor-General was dismissed, a vacancy would then occur, just as it does when a Governor-General resigns (this has happened 3 times). Section 4 of the Australian Constitution allows the King to appoint an administrator to carry out the role of Governor-General when there is a vacancy. The longest-serving state governor would take on the role of Governor-General until the position is permanently filled.

The Governor-General engaging with the Australian community.
The Office of the Official Secretary to the Governor-General
The Governor-General engaging with the Australian community.

The Office of the Official Secretary to the Governor-General
Description
The Governor-General seated at a large table talking to community members.