David Barry Conway was appointed Assistant-Surgeon Royal Navy on 5 October 1811. He was employed as Hospital Mate at the Royal Hospital Haslar in 1815. 
He was promoted to Surgeon on 5 October 1822 and appointed to the Harrier in 1824 
David Conway was employed as Surgeon Superintendent on two voyages to Australia:
1). The Manlius which departed the Downs 17 April 1827 and arrived in NSW 11 August 1827. He kept a Medical Journal from 23 February to 24 August 1827.
The surgeon remarked that diarrhoea and fever were the most prevalent diseases on the voyage out. The ship experienced very bad weather in the Latitude of the Cape and from there to Sydney there were gales and heavy seas which from it breaking so frequently on board kept the ship very wet. The Prisoners, when the weather was fine, were all on deck and the greatest attention to ventilation and moving the stoves with fires about the prisons was paid and the decks were scraped dry. A great many prisoners were diseased having suffered much in the hulks. Two convicts died on the passage out - Richard Mansfold and William Wheeler. 
2). The Georgiana which departed Plymouth 15 December 1828 and arrived in Van Diemen's Land on 20 April 1829.
On 2 May 1829 The Hobart Town Courier described some of the convicts of the Georgiana -
We regret to observe so many of the prisoners by the Georgiana, consisting of mere boys, on an average not more than 10 or 12 years of age. Their youth is certainly a fault that time will improve, but in the mean time it must be very distressing to the Government to know how to dispose of them with propriety. Some of the elder may indeed soon learn to officiate as bullock drivers along with the ploughman, or even as hut keepers and cooks at the stock runs, but the majority are we fear incapable of even such service. The great scarcity of labourers at present in the island, however, must in a great measure speedily relieve the government from the care of them, for we can conceive no plan worse than allowing them to remain in the barracks at Hobart town where they very worst examples must be incessantly, before their eyes.
We remarked one little fellow among them not much more than 4ft high and about 10 years old, who has been in prison nearly 4 years under conviction. When asked by the Principal Superintendent how old he was, the little urchin answered 'he was so young when he was born that he could not tell'. His name we believe is William Edwards, but he is generally known by the appellation of King John. He is one of those unfortunate instruments of the old thieves with which London, notwithstanding all our weeding, still superabounds, that used to be carried in trunks or boxes and left at houses, or covered up in a basket with cabbages, etc. and placed in a convenience corner until night, when it was his duty to open the street door for his confederates to enter or sometimes he was thrust in at a cut out pane of a shop window, which he would afterwards strip.
Not more than 3 of all the three score boys on board the Georgiana could repeat even the Lord's prayer at the departure of the vessel from England, but now we have much pleasure n stating owing to the persevering and praiseworthy exertions of the Surgeon Superintendent Dr. Conway, they can not only all repeat their prayers but most of them their catechism.
It is to be hoped that the work of reformation which has been so well begun will advance and be perfected in these boys by their removal to this island. His Excellency on Tuesdays morning, in the course of his usual address to these men, on their being assigned to their different employments, could not help remarking the disgraceful levity on the countenance of some of them, at the very time they ought to have been covered with abject shame, and a sense of the disgrace attached to banishment, for their offences from their native country. Many others among them, however, we are happy to say evinced a contrite temper and a firm resolve to merit the indulgences held out by government to the sober and honest labourer
In 1832 the Hampshire Advertiser reported on 23 June 1832 that David Barry Conway, surgeon of the Ordinary at Chatham had died after a few hours illness. 
His widow Ellen died over fifty years later on 2 November 1888 at 7 West end Terrace, Winchester. Ellen was the daughter of (the late) William Day, Post Captain R.N., formerly Governor of Sierra Leone.