There?s a Moses Mulholland listed in Griffiths Valuation for 1859. He was farming plot 13, in Derrygortanea, in the parish of Desertcreat, Tyrone. That was a farmhouse, outbuildings and c 18 acres. He also ha da share of some land on plot 12. That was the only Moses M in the county in Griffiths. (Not everyone was in Griffiths). The revaluation records show him leaving around 1862/3, so that fits with your information.
That farm today appears to be on the Thornhill Rd, between Cookstown & Pomeroy.
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=nameSearch
There?s a marriage between Moses Mulholland and Catherine McMinn registered in Dungannon on 5th Nov 1847. You can view a copy of that marriage cert on-line for ?2 on the GRONI site. The townlands (addresses) would confirm whether it?s the same family as lived at Derrygortanea.
https://geni.nidirect.gov.uk
The births and deaths are prior to the start of statutory registration (1864) and so you would need to rely on baptism records for the births. Unless the family were Church of Ireland you won?t find any death records as other denominations generally didn?t keep any. A gravestone?s probably the only other possibility.
