At the very least there should be a publicly accessible model of population growth to verify the claims being made in this debate. The productivity commission has modelled population growth for this purpose, although the intricacies of the model are not at all clear or public.
My first step towards this is to actually look at the historical demographic record in Australia. As you can see from the interactive chart below, the country's age dependency ratio has been steadily increasing since at least the 1970s.
Offsetting this age dependency has been a quite dramatic decline in youth dependency as fertility rates fall. The total dependency, or number of children and retirement age people per working age person is at record lows.
The next step is the add some features to this model to allow a choice of assumptions about immigration rates (and ages) as well as birth and death rates, to see exactly what how immigration policy is affecting demographics and whether there are some circumstances in which the 'immigration solves ageing' slogan may hold.