Try searching for him and Annie in the Australian records on Familysearch (as Gadget has posted while I was typing) which suggest he died in Rutherford in 1920. Also try Trove newspapers (see below).
This is, I think, Annie's death notice" Newcastle Morning Herald" Sat 27 Jan 1940(nb Newcastle in NSW!)
On18th Jan at her daughter's residence, Hall Street,Heddan Greta,Ann,relict ofthe late George Lishmanof East Greta,dearlybeloved mother of Bella (Mrs J Hind,England),Winnie ( Mrs A Stagg,Tamworth), Maggie (Mrs R Logan,Heddon Greta) Eva ( Mrs A Watt,Lambton) age 91 years.
And on Sat 5th Feb 1821 (same paper)
In Memoriam
A tribute of love to the memory of our dear husband and father George Lishman who passed away at East Greta 5th Feb 1920.... (Inserted by his loving wife Annie and daughters Bella (England), Winnie,Maggie and Eva.
Looks as though it was George,Annie, Winifred and son John who emigrated toAustralia(where two further daughters were born,Margaret and Eva). Isabella has stayed in England and married a Mr Hind.
William