I can only tell you the legal status that India as a Dominion had between 15th of August 1947 and 26th of Jan 1950 (the day India's Constitution came into affect) As per the Independence of India Act 1947 India became a Dominion of Britain this can be said as a transition period between the formulation of the Indian Constitution by the Constituent Assembly and its enactment. India in this period was administered by an Amendment to the Government of India Act 1935 which provided an Interim Constitution for the said period.

The provisions of the Act were:

Abolition of the Sovereignty and Responsibility of the British Parliament:

As per the Government of India Act 1858 the British parliament became the guardian of India for the English Monarch. An office of the Secretary of State for India(a member of the British Cabinet) was created to manage the affairs of India.

The Independence of India Act removed this subordinate status of the Government of India to the British Government and the suzerainty of the British Crown over the Princely States and the relations with the Tribal areas also lapsed. Thus the Crown was no longer the source of authority as it had been under the federal scheme as per the Government of India Act 1835.

Governor General and Provincial Governors to act as Constitutional Head:

This is the provision because whcih it can be said that India was the Dominion of the Britian as its Constitutional Head was quintessentially an Officer of the British Crown. The Governor General however was a mere figure(though Lord Mountbatten did an exceptional job when it came to using his influence as the Governor General esp. in case of the Integration of the Princely States into India) and had very little power over the Legislature and the Executive.

Sovereignty of the Dominion Legislature:

The Central Legislature(The key law making body of British India) which consisted of the Legislative Assembly and Council of States was abolished and the power of the Central Legislature were given to the Constituent Assembly of each of the two Dominions. Thus the Constituent Assembly had the Dual function of Legislation and Constituting(Though no major bills were passed by this Assembly and as witnessed in the debates on the controversial Hindu Code Bill that several members were dead against the formulation of key legislature by the Assembly as its members were not elected directly by the People of India).

I have no Idea about how the Dominions in Australia work so if hope that somebody would post it so that a comparative analysis could be done.

Source:

Introduction to the Constitution of India by Dr. D.D.Basu(1960)

India after Gandhi by Ramchandra Guha (2007)(just that minor part regarding the debates on the Hindu Code Bill)