The Earl Grey transported convicts to New South Wales in 1836 and 1838 and to Van Diemen's Land in 1842.
The prisoners of the Earl Grey were convicted in counties in Ireland - Limerick, Kerry, Tipperary, Cork, Roscommon, Waterford, Longford, Mayo, Queens, Kildare, Dublin, Galway, Antrim, Clare, Cavan, Carlow, Wicklow, Westmeath, Down, Leitrim and Kilkenny. There were also eight men who joined the ship at the Cape of Good Hope.
There were farm servants and labourers, blacksmiths, boatmen, haberdashers, carters, gardeners, poulterers, indoor servants, grooms, coachmen, calico printers, errand boys, fishermen, a cooper, leather stainer, polstillion, linen weaver, sawyer, tobacco spinner, biscuit maker, gamekeeper, a cordwainer, ivory bone cutter, soap boiler, butchers, merchant's clerks, attorney's clerk and one prisoners, James French who gave his occupation as 'gentleman'.
Their crimes included various forms of stealing and robbery, receiving, coining, embezzlement, perjury, forgery, manslaughter, assault, murder, rape desertion, striking an officer. Quite a few had been sent for Whiteboy crimes.
Surgeon William Evans
William Evans kept a Medical Journal from 15 July 1836 to 5 January 1837 during the voyage from Deptford, Kingstown and Cork to Sydney. He joined the Earl Grey on 15 July 1836 at Deptford
On 25th July, the soldiers of the Guard joined the vessel. The Guard consisted of Lieut. Ronald McDonald of the 80th regt., Lieut. R. B. Hill, 41st regt., Quarter Master Potter 4th regt., Assistant Surgeon Graydon 50th regt., Assistant Surgeon Allman 4th regt., one Sergeant and 29 rank and file of the 28th, 50th and 80th regts., 5 women and 7 children. 
Cabin Passengers included Mrs. Macdonald, Mrs. Allman and Mrs. Potter.
Deptford To Kingstown
They sailed from Deptford on the 27th July and reached Kingstown on the 14th August. On 16th August, 91 convicts were received from the Essex Hulk.
The following day the Earl Grey sailed for Cork arriving there on 21st August. On the 23 August, 192 convicts and 5 free boys, the sons of convicts in the colony were embarked. In total there were 384 people on board.
The Earl Grey departed Cork on 27 August 1836.
Cork and the latitude of Madeira there were a few slight cases of fever arising from the damp, crowded dormitory. After passing the Cape Verde Islands the heat became oppressive and on entering the rainy regions thirteen men were suffering from scurvy. William Evans thought scurvy and scorbutic dysentery seemed to arise partly from the impurity of the water but also from foul stagnant air between decks, combined with depression, anxiety of incarceration and sea diet. The water had been taken on at Deptford with very little care and sometimes at improper times of the tide, though Government regulations were imperative on this head.
Crossing The Equator
On the evening of 1 October they crossed the equator with a fine breeze from south southeast and passed rapidly through the south east trades, reaching the Tropic of Capricorn in 11 days from the line. There were now 30 men suffering from scurvy and on 19th October William Evans recommended the master to call at the Cape for refreshment, experience having taught him that 'lime juice and nitrate of potash are mere prophylactics' and that fresh meat and vegetables were the only sure means of ensuring health for the rest of the voyage.
Cape Of Good Hope
At 5pm on 4 November the Earl Grey anchored in Simon's Bay, where they remained for eight days. They received fresh beef, mutton and vegetables and took on board 5 live bullocks and 60 sheep. In less than a fortnight, thirty men who had been bed ridden were convalescent and continued to improve in spite of the weather.
Eight prisoners were taken on at the Cape - Jacob alias Jacob Asolie; George Hay Edward Henry Elphie James English Joseph Job Broadbent Robert Wicks Jan Windvogel David Williams
Although he had not had the chance to examine the convicts, Surgeon Bailey R.N., at the Cape of Good Hope referred to the prevalence of scurvy on the Earl Grey in Memorandum on Scurvey by Surgeon Bailey R.N. Superintendent of the Somerset (Civil) Hospital at the Cape of Good Hope. On 20th December 1836, Cape Otway was sighted and soon after the north end of King's Island. The following day they passed through Bass Straits with a fine breeze from the westwards.
On 31 December 1836 the Earl Grey reached Sydney after a voyage of 18 weeks from Cove of Cork. The weather was fine and by the time they reached Sydney. The convict indents report three deaths during the voyage - Charles McCarthy, Michael Coyle and Patrick Bryan.
The Military Guard were landed and marched to their barracks on the 2nd January 1837.  The convicts remained on board where a muster would have taken place. The indents include name, age, education, religion, marital status, family, native place, occupation, offence, date and place of trial, prior convictions and physical description. There are occasional notes about dates of death, pardons and tickets of leave but no details of where the men were assigned. 
They were landed 'at the point' on 18th January 1837. According to the surgeon, there was not one of the 288 convicts who were landed who could not walk to the Hyde Park Convict Barracks to be inspected.
Notes and Links
1). Cork County Assizes - Michael Twomey, who was convicted of the manslaughter of his wife at Passage, after a brief but impressive address was sentenced to transportation for life. - Freeman's Journal 4th August 1836
2). Kildare Assizes - A fellow named William Nowlan was indicted for perjury, in having sworn informations against several of the country people, respecting the manufacture of pikes and the tendering of unlawful oaths. He was found guilty and sentenced to seven years' transportation. - Freeman's Journal 13 July 1836