Hi, I was just wondering whether you've found some cuttings yet? My parents have a huge Mulberry tree in their backyard and I'll be able to take cuttings for you.
Where in Australia are you?
It's probably best if I wait until spring to get any cuttings.
Cheers

I wouldn't wait until spring to take cuttings from a black mulberry, as the tree should be dormant. Here's what the Qld DPI website says:
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Mulberry trees can be grown from cuttings collected in winter. The cuttings, taken from strong 1- or 2-year-old wood, should be about 30 cm long. They can be struck in a plastic pot containing a 1:1 sand-peat mixture, or planted directly into the place where they are to be grown.
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This is what a Californian website says:
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Mulberries can be grown from seed, although the plants can take 10 years or more to bear. Seed should be sown as soon as extracted from the fruit, although white mulberry seeds germinate better after stratifying one to three months before planting.

Sprig budding is the most common method for grafting mulberries. A T-cut is made in the rootstock and a smooth, sloping cut is made on the lower end of the scion. The scion is then inserted into the T and wrapped and sealed. Other types of grafts are also usually successful, although there may be incompatibility between white and black mulberries.

Hardwood, softwood and root cuttings also are suitable methods for propagating mulberries. Softwood cuttings of white mulberries root easily when taken in midsummer and treated with rooting hormone. Red mulberries are less easily rooted. Black mulberries are also somewhat difficult to propagate since they tend to bleed a lot.
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So for black mulberries, the best time to take cuttings is now.