Mourney Family - Which county?

Hello!

Henry Mourney settled in Scotland from Ireland around in the early 1800s, I think. He was born in Ireland c. 1784, but we have no idea which county or parish he came from. He and all his descendants were sailors, sail-makers and ship's captains in Troon/Dundonald in Ayrshire, Scotland, so the chances are that he came from a seafaring region in Ireland. The name, Mourney, is possibly a hint, with a connection to the Mourne mountains, but there is nothing more concrete than that to indicate a link - but if the family is from the Mourne region, would this be County Down? In Scotland the family were all protestants, but that might not mean much - he could still have been a catholic, I guess, until he married the Scottish lassie (Mary Hewitson, b. 1782). From my internet research, it would seem that there are very few Mourneys still aroundanywhere in Ireland - but there are records of some migrating to America and Australia directly from Ireland. Henry Mourney's Scottish descendants were quite plentiful for a while in Ayrshire, but they seem to have all immigrated or died out from there, too. I am a descendant of Henry's grandson, James (1848-1919), a sail-maker, who immigrated to Sydney, via New Zealand, c. 1875. I would be very grateful of any suggestions of where to start looking - for example, was there a town or county with a tradition of trade or immigration with Ayrshire?

Thanks,

Kristine Mourney (from Melbourne, but currently living in Germany).