Like you I have been unable to locate the David Dorans and Jean Duffy in question. Perhaps one lead to follow is through the maternal grandparents of Lizzie Dorans in the 1911 Census. What are their names and ages at that date? I have discovered that the children of David Dorans in the Kilmarnock Census of 1881 apparently emigrated to America, and he may have gone with them, as I have found no death record for him. Their names were Agnes b 1852, Patrick b 1856, and twin brothers John and Thomas b 1863 and at least 3 of them can be found on the FamilySearch.org website to be in the USA (I am not sure about Thomas who may have remained in Scotland).
Where are you located yourself, i.e which continent? I am to be found in Toronto, Canada.