Mary,
Catherine & James births are well before the start of statutory birth registration in Ireland (1864). To trace them you would need to know their denomination and likely parish in Ireland. But there aren’t that many churches with records for the 1700s so there’s a strong chance that even if you can find where they come from, there may be no records to consult.
McConachy (and it’s many variant spellings) is very common in Co. Antrim. Looking at the 1901 census there were 333, of every denomination, though the majority were Presbyterian. (So they originated in Scotland, most probably arriving in Ireland in the late 1500s or early 1600s). A very large percentage of the population in north east Antrim were originally from the McDonald Estates in Scotland (Islay, Jura, Gigha, Kintyre etc), arriving when Sorley Boy McDonnell came from Islay, with thousands of McDonald/McDonnell tenants, to establish the clan in Antrim in the mid 1500s.
My experience of previous research on folk from Antrim who ended up in Campbeltown in the 1800s is that they tended to come from coastal communities, so places like Ballycastle, Glenarm, Cushendun & Cushendall. (Cushendun had a ferry service to Scotland from the 1500s, if not before). Until the Antrim coast road was built in Victorian times, connecting it to Larne and beyond, the north Antrim coast was cut off from most of the rest of Ireland and so it was easier to do business in Scotland than in Belfast or Dublin. As you probably know, it is only 13 miles across to the tip of the Kintyre peninsula and folk were back and forth all the time. That’s the area I would be inclined to focus on, but there are very few church records from the 1700s. If any at all. DNA testing might be the only way of linking in to families in that area today.
