Irregular marriages were entered into by persons who took advantage of the principle in Scots Law that marriage was constituted by mutual consent. The form of contract generally adopted was a simple written declaration of acceptance of each other as husband and wife. This was not the case in England and Wales and led to a thriving marriage business just within the Scottish Border (notably at Gretna Green) for English couples who eloped together to marry in Scotland. They were married by "priests" who would provide a witness and enter the marriage in a register. These registers are not held centrally and indeed the whereabouts of some is unknown.
This pdf copy of the leaflet Irregular Border and Scottish Runaway Marriages (8th edition, 2014) compiled by Ronald Nicholson provides details of the extant registers and their current (or last known) location. It is made available on our website by kind permission of the author.