The tithe applotment records in Ireland generally list farmers. The records for Mullaghbrack (1834) list 2 Robert Livingstons. One farmed in Carrickleany and the other Coolmillish. (The 2 townlands share a common border). Carrickleany appears to be an alternative name for Carricklane which is actually in Kilclooney parish.
http://www.irishgenealogyhub.com/armagh/tithe-applotments/mullaghbrack-parish.php
http://www.irishgenealogyhub.com/armagh/tithe-applotments/kilclooney-parish.php
(Those records also list William Reid in Marlacoo Beg).
Griffiths Valuation has a Robert Levingston in Carricklane. He had plot 5 which was a 17 acre farm. No linen mills. He may have frown flax and his family and their labourers may have made some linen but it would have been hand woven on portable machines kept in the home, such as are still used today to make Harris Tweed in the Outer Hebrides. There was a mill pond in the townland but I suspect that was for a corn mill. No sign of where it actually was though. It isn’t listed on Griffiths and I suspect was gone by the 1860s. Some of the property is still farmland today (on the modern Seaboughan Rd). I think there’s a factory unit where the farmhouse once stood.
The Valuation revision records are noted that the farm passed from Robert to Alexander in 1868. (That’s the year the Griffiths clerks were aware of it, the event may have been a year or two earlier.). The records are noted “Robert dead.” Copy attached.
There is a death in the Armagh registration area for a Robert Livingston, aged 90, on 15 Mar 1864, regd Armagh. Might be him. It’ll cost you £2.50 (sterling) to view on the GRONI site.
Alexander Livingston looks to have been his son and was married to Sarah Mary Fullerton. Here’s the birth of their daughter Sarah Mary in 1864:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1864/03619/2335636.pdf
Alexander & Sarah appear to have married pre 1845 and so their marriage doesn’t appear in the civil records.
Alexander died on 22.5.1902 aged 80. His will is on-line on the PRONI site and mentions son Robert and daughters Sarah Mary & Agnes. Family were Presbyterian and so may have attended one of the 2 Presbyterian churches in Markethill. Son Robert inherited the farm. He died 30.10.1922 and it then went to his wife Margaret.
Portadown News of 31st July 1954 reported that Alphonsus Livingstone of Carricklane, Markethill was fined £2 for driving without due care and attention.
Same paper on 2nd Sept 1932 reported that James Livingston of Carricklane had been fined5 shillings for not having his children vaccinated.
Same paper on 27th April 1951 reported that William Livingstone of Carricklane, Markethill was fined £1 for allowing 9 sheep to wander on February 27th.
Belfast Telegraph of 19th Aug 1925 reported that Mary Livingstone of Markethill had been awarded a scholarship to attend a local secondary school.
PRONI has a record of a Robert Leviston of Carricklane having a vote (in parliamentary elections) in 1815. ARM 5/2/3. Leviston looks to be a variant of Livingstone. He voted for the Earl of Gosford (his landlord). Smart move. He was a freeholder which means he had property worth over £20 in rateable terms, and he had a lease for life. (So quite well off by the standards of the time).
D1606/3/9 at PRONI is a list of leases for Carricklane 1783 – 1828.
Probably worth searching the Registry of Deeds records for Carricklane on the LDS site. You might get a 3 lives lease for The Livingstones of Carricklane which can often given you another generation or so. There were a lot of them granted in the 1780s and 1790s.
One of the witnesses to the 1853 marriage was Adam Reid. This might be him marrying Mary Ann Brown in 1846, in which case he looks like James Reid's brother:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1846/09300/5367285.pdf
Adam Reid lived in Cabragh. Griffiths lists him there on plot 31 which was just under 5 acres.
Markethill 1st Presbyterian church has records starting in 1843; Markethill 2nd has records from 1821. Copies of both sets of records are held in PRONI in Belfast. Assuming either were the churches the families attended you will obviously struggle to get back any earlier, due to the lack of records.
Being Presbyterian, having Scottish surnames (Livingston & Reid) and living in Armagh, all tend to point pretty strongly to Scottish origins. The families are likely to have arrived in the area in the 1600s, perhaps as part of the Plantation. (No Livingstons in Armagh in the Muster Rolls c 1630 so they apparently arrived after that but there were some Reeds so they may have been there since the early 1600s).