Not sure if this refers to your peopleCalendar of Wills1868Andrew Moore - late of Dagmar Street Belfast , Gentleman, a Widower died 29 May 1868Letters of Administration granted to James Moore, Beersbridge Cottage, Ballymacarrett, CountyDown, Flax Merchant, one of the next of kin.
The words 'Lived Private' (ie. private means / income, there are lots of examples of 'living private' stated for occupation in the 1901 and 1911 Ireland censuses) would indicate that Andrew Moore was deceased. I also wondered about the following local newspaper death notice in April 1874: 'MOORE - April 10, at Randalstown, Nancy, relict of the late Mr. Andrew Moore, aged 66 years.'. Relating to that notice (albeit death date is out by a day), there is the following gravestone inscription in Drummaul Church of Ireland graveyard: 'Erected by George Moore, in loving memory of his father, Andrew Moore, died, 5th February 1863, also his mother, Annie Moore, died, 11th April 1874'. Just a possibility.