For information on registers (baptisms, marriages and burials) for a particular parish, please see that parish's page. General advice on parish registers throughout Scotland can be found under Church Records on the main Scotland page in GENUKI.
The website of the National Records of Scotland includes a leaflet on irregular marriages and information on the known surviving registers. Irregular marriages occurred along the Border and were a form of marriage by consent, convenient both for English runaway couples and Scottish Borderers who did not want to marry in their own churches. The Church of Scotland disapproved of such marriages and would often catch up with a couple, perhaps when their first child was born or baptized. So kirk session minutes can be another useful source for tracing irregular marriages.
Graham and Emma Maxwell are transcribing and indexing baptisms, marriages and burials recorded in Scottish Borders kirk session records and non-conformist church records. Their website also offers a free search facility for these resources.
The kirk session of a parish consists of the minister of the parish and the elders of the congregation. It looks after the general wellbeing of the congregation and, particularly in centuries past, parochial discipline. Most kirk session records are held in the National Records of Scotland in Edinburgh and can be fascinating reading. For more information see Anne Gordon's Candie for the Foundling published by the Pentland Press in 1992. ISBN 1 872795 75 7 (720 pages).
For an account of the Border kirk session records, focusing particularly on poor relief and the dispensation of discipline, see M.C. Lawson's article "The Poor, Crime and Punishment, and the Power of the Kirk in the Borders, 17th & 18th Centuries" which was published on pages 14-15 of the June 1996 Borders Family History Society magazine.
See also sections on kirk session records of Bunkle and Coldingham parishes.
The LDS Family History Library catalogue lists a microfilm version of Presbytery meeting minutes, 1843-1862 of the Free Church of Scotland's United Presbytery of Duns and Chirnside. The catalogue entry records this as a microfilm of records held at the Scottish Record Office (now called the National Records of Scotland), citing reference CH3/104/1. The microfilm copy in the LDS catalogue should hopefully be viewable at LDS family history centres around the world.