Family of Alexander Cairens
Alexander Cairens married Mary Binnington on 10th Feb 1779 at Market Weighton (although the IGI index say it was at Shipton by Thorpe). In the marriage banns he is described as "of the Northumberland militia, quartered in this parish", but in the marriage record he is described as a labourer. (So banns of marriage can give useful information).

Here are details of his earlier life in the Northumberland militia and discusssion of where he came from.
They had 3 children :-
Sadly, Mary died and was buried on 18th June 1788 in Market Weighton.
Alexander then found a second wife Mary Parkin, who he married on 23rd Feb 1789 in Cottingham near Hull. They had another 10 children, all baptised in Market Weighton :-
Jane Ann Brigham bap 1813, married Richard Dry, child Francis Henry Dry, descendant is Francis Riley in New Zealand Eliza Brigham baptised 1814 Robert Brigham baptised 1815, Game Watcher, married Mary Ann Brown James Brigham baptised 1817, Cordwainer, married Mary Gore. Had at least 5 children including Henry Cains Brigham. Hannah Brigham baptised 1819 Caroline Brigham baptised 1820
Their father Alexander died and was buried in Market Weighton on 14.1.1823, aged 67.
My family called themselves Cains after about 1810. Cains is still quite an unusual surname, so I have tried to trace children of each of these using the IGI registers, as shown above, with various spelling of the surname, but I can never be certain that a person marrying is the same as one of Alexander's children. (The Cairens family are listed in the IGI index, in the Karren section, but they have details wrong, such as having Alexander's wife as Anne).
As far as I knew, my two brothers and I were almost the only Cains descendants of all this family because many people had daughters who lost the Cains name on marriage, but I have recently had contacts from lots of Cains relatives in Hertfordshire and Australia. Please email me if you know of any other descendants of these people, especially still living. My email address is simon @ cains . myzen . co .uk
One book about the Cairns family of Scotland has been published :- A History of the Cairnes or Cairns and its connections, by Henry Cairns Lawlor, published by Ellliot Stock 1906. It is very hard to track down. There is a digital scan here http://archive.org/stream/historyoffamilyo00lond/historyoffamilyo00lond_djvu.txt with the usual text recognition errors, and doesn't include the family trees from the book.
In Burke's Peerage, there is a Cairns family, but the oldest Cairns family are detailed in the "Rossmore" family entry.