John Milbank Garity (many spellings of this surname) was born on the 7th June 1826 onboard The Chapman. The Chapman was a convict ship which left Woolwich, London, on 4 April 1826, with 100 convicts, and a crew of sailors and members of the Royal Staff Corps. The ship arrived in Hobart on 11 October 1826, and left for Port Jackson on 22 October reaching NSW on 4 November 1826.

The doctor's log of the voyage refers to Private Charles Garity RSC [Royal Staff Corp] who was being treated by the doctor for a sore throat. In addition, on June 7, the doctor notes that Ann Garity, "wife of Private Charles Garity", had given birth to 'John Milbank Garity'. Note that John Milbank was the Master of the Chapman. 

Charles Garity stayed in Tasmania with other RSC soldiers and may have worked on roads & bridges in the newly established town of Oatlands. I suspect he died in the early 1830s.

It is certain that John along with sister Ellen and brother William were abandoned by parents. This is the text of a memorandum from Lt Gov Sir George Arthur to the local Colonial Secretary John Burnett.

Memorandum
28th Sep

Be so good as to direct that John Garrity 7 years of age son of Charles Garrity formerly a soldier in the Staff Corps be received into the Orphan School this day. The Child is perfectly destitute, his cruel & unnatural Parents having totally abandoned him.

GA
The Colonial Secretary

Urgent

Prepare the necessary directions as quickly as possible. The Boy is now waiting for admission.
JB 28 Sep

As for John, I have found one possible "sighting" in Tasmania.

An advertisement in the Austral-Asiatic Review, Tasmanian and Australian Advertiser of Tuesday 16 February 1841, indicates that a John Garraty aged 15 absconded from an employer who he had been working for, for 3 years. (Presumably while living at the orphanage for much of this time.) I think it reasonable to think we are seeing the John Garrity born on the Chapman here given the perfect matching of his age.

John died on 14 August 1888 at the Bourke St Model Lodging House, Melbourne, Victoria. This was a home for poor single men. He had been a jockey for years in Victoria.

John Garrity is my great great great uncle.