The Plantation is a subject which arouses a lot of strong feelings. I am not an expert, and wouldn’t claim to have any authoritative knowledge. If you want to read more detailed accounts by experts, I’d suggest:
Eagles Wings by Dr David Hume
The Planation of Ulster by Philip Robinson
The Scottish Migration to Ulster in the reign of King James I by M Perceval-Maxwell
But my understanding of the broad historical position is that in the 1500s Britain and Ireland had been the subject of an attempted invasion by Spain (the Spanish Armada). Bad weather and a bit of tactical smart behaviour by the Royal Navy had seen them off, but there was a real fear that they would return and that also the French would likewise invade. Ireland was seen as an easy back door into the British Isles. If you could land an army in Ireland, it’d be easy to move on to England. (A bit like the Cuban missile crisis in the 1960s, as far as the US was concerned). So government strategy was to ensure that Ireland was not somewhere that it would be easy to land a foreign army. Naturally most of the native Irish were not very pro-British so it was decided that the answer was to swamp Ireland with British (ie Scots, English & Welsh) settlers, so altering the dynamic of the country to ensure that it became more pro British and more able to resist any invasion. In the final years of Queen Elizabeth 1st’s reign there was a stand off with some of the big Irish landowners. This led to a couple of battles and what was called “The flight of the Earls”. 2 Earls and about 90 kinsmen fled Ireland (for Spain) in 1607. Their lands were declared forfeit. See:
https://en.wikipedia.org/wiki/Flight_of_the_Earls
Elizabeth I had no children and as a result the king of Scotland (James VI of Scotland) was invited to rule both Scotland and England. He was known thereafter as James I. His strategy was to invite Irish landowners to sign a loyalty oath to the Crown. If they did that (and many did, Ulster examples being the Maguires in Fermanagh, O’Kanes in Co Derry and O’Neills in Co Antrim) then they generally kept their lands. Some did not, and their lands were therefore declared forfeit, as were also the lands owned by the Earls who had fled.
So in the period 1600-1610 you had a situation where some native Irish kept their lands but in many other places new landlords arrived from Scotland, England & Wales to occupy lands declared forfeit. Their right to occupy the lands came from the king, to whom, all land ultimately belonged anyway. Even today that right exists (in the UK) to the extent that if someone dies intestate and no eligible relative can be found, the property goes back to the Crown on the basis that it originally held the title anyway.
But to answer your question about how the landlords got their land and what happened after that, the answer is that some were native Irish who retained their lands (about 30%) and the others were settlers from Britain. Once they had taken title to the land, under royal warrant, their position was exactly the same as any landlord anywhere. Same as today. They could sublet, sell or set leases on any terms they chose.
The general arrangement for the Plantation was that the King invited big landowners in Britain to go to Ireland. He granted them land with low rents on condition that they developed the area. So they undertook to build public buildings, a bawn (defensive fort) and roads etc. They were therefore called “undertakers”. They also had to undertake to take a significant number of their tenants with them from their estates in England, Wales or Scotland to ensure there were enough people to make the new settlements viable. (Previous attempts to settle Ireland had failed due to lack of incomers). So many of the first “ordinary” settlers came from the parts of Britain that their main landlord had properties.
The King wanted people he could rely on. Though he was King of all the UK, James I was Scottish and he favoured Scots. He encouraged a lot to go to Ireland because he felt he could rely on them.
Ardara is in the barony of Banagh. In the early 1600s, that was settled by landowners from Ayrshire, Wigtownshire, and Kirkcudbrightshire in Scotland. So possibly that’s where your ancestors originated. See:
http://orrnamestudy.com/plantnire1610.htm
According to Perceval-Maxwell, Boylagh & Banagh Barony was settled by Sir Robert of Bomby from south Kirkcudbrightshire.
Apart from the names of the main undertakers, no names exist of individual tenants and artisans and so it’s not normally possible to trace them back to Scotland via paper records, though you might get lucky with DNA.
It’s seems to me from the information that Cordangan has provided that the Connolly family were native Irish who acquired lands in Banagh barony in the late 1600s. Presumably sold to them by the previous owners/tenants (possibly Sir Robert of Bomby but I don’t know enough about the history of the area to say for certain).