Exactly what will happen depends on the structure of the relevant company.
For example, if it is a sole director and sole shareholder company there is a section of the Corporations Act that provides a pathway (extracted below, section 201F).
Similarly, if it is a sole director and not a sole shareholder company, generally it is the case that the shareholders will have the ability to appoint new directors pursuant to another section of the Corporations Act (again extracted below, section 201G).
There are additional issues to be aware of where the company acts as trustee of a trust (for example, any appointor powers) or trustee of a SMSF (for example, the requirements under the superannuation laws);
Having said the above, the conservative approach would be to create powers of attorney now for each company, although this approach does obviously involve additional costs and complexity for each company and ongoing compliance issues.
If a person who is the only director and the only shareholder of a proprietary company:
(a) dies; or
(b) cannot manage the company because of the person’s mental incapacity;
and a personal representative or trustee is appointed to administer the person’s estate or property, the personal representative or trustee may appoint a person as the director of the company.
A company may appoint a person as a director by resolution passed in general meeting.