Greg,
There aren’t many records for Co. Armagh in the early 1800s. Statutory registration didn't start until 1864. Prior to that you have to rely on church records. Not all parishes have records and of those that do, not all are on-line. So it can be difficult.
You haven’t said what denomination your family were.
The tithe applotment records list those with land. So servants, labourers and others without land aren’t listed but farmers usually are. If your ancestors were farmers, then they could be in the tithes. Here’s a list of Thorntons in Armagh around 1830, so potentially useful areas to focus on:
Thornton, Alexander-- Townland : Artabrackey Year : 1828-Drumcree-Armagh
Thornton, Anne-- Townland: Toneyvalton Year: 1833-Kilmore-Armagh
Thornton, David-- Townland : Drumnakelly Year : 1828-Drumcree-Armagh
Thornton, George-Townland: Lurgan Town Year: 1833-Shankill-Armagh
Thornton, George-Townland: Tullyglush Nevin Year: 1827-Tynan-Armagh
Thornton, Henry-- Townland : Artabrackey Year : 1828-Drumcree-Armagh
Thornton, Henry-- Townland : Ballyworkin Year : 1828-Drumcree-Armagh
Thornton, James-- Townland: Clonlum Year: 1834-Killevy-Armagh
Thornton, James-- Townland: Clonlum Year: 1834-Killyman-Armagh
Thornton, John-- Townland : Drumgreenagh Year : 1825-Derrynoose-Armagh
Thornton, John-- Townland : Foy More Year : 1828-Drumcree-Armagh
Thornton, John-- Townland: Clonlum Year: 1834-Killyman-Armagh
Thornton, John-- Townland: Clonlum Year: 1835-Killevy-Armagh
Thornton, John-- Townland: Toneyvalton Year: 1833-Kilmore-Armagh
Thornton, Major - Townland: Tullyard Year: 1832--Grange-Armagh
Thornton, Michl.-- Townland : Rassan Year : 1828-Creggan-Armagh
Thornton, Neal-- Townland: Clonlum Year: 1834-Killevy-Armagh
Thornton, Neal-- Townland: Clonlum Year: 1835-Killyman-Armagh
Thornton, Owen-- Townland : Madden Year : 1825-Derrynoose-Armagh
Thornton, Philip-- Townland : Rassan Year : 1828-Creggan-Armagh
Thornton, R.-- Townland: Legarhill Year: 1832-Grange-Armagh
Thornton, R.-- Townland: Mullynure Year: 1832-Grange-Armagh
Thornton, William-- Townland : Fernaloy Year : 1825-Derrynoose-Armagh
Have you been able to find marriage or death records for James & Richard? If so, do they contain details of their parents? That would be very helpful to try and trace the families.
Markethill is split between the parishes of Mullaghbrack & Kilclooney. If you know the Thorntons denomination, I’ll advise what parish records exist.
Researching in Ireland in the 1700s and early 1800s is very hard going due to the general lack of records. If you don’t know where they lived it’s a needle in a haystack. Ideally you need to know the person’s exact denomination and the townland or parish they lived in to have any chance of finding them, and even then there may not be any records for that location.
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).
