You haven't given any dates for your family members.However, I do have one "Duke" recorded.I have no information on his descendants so am unable to ascertain if he is the one you are looking for.The information I have comes mainly from Burke's Landed Gentry of Ireland, with other internet sources adding to the data.
"Duke"'s great-grandfather was Paul of Gortner Abbey and Killiney co. Mayo, Commander R.N. 1762 d.1786 Crossmolina = Mary O’MALLEY (of George of Hurrisk/Murrisk. They had 5 sons, the first named being Paul = Rebecca Caldwell.
The second son named was: William Ormsby b.1765 d.1849 = Mary Cooper (dau of James Cooper Esq of Annagh , inspector of excise co. Mayo& Miss Kirkwood [of William of Glenora co. Mayo])
William and Mary had one son recorded, namely: 1]- William Henry Ormsby = Eliza (of Walter Bourke)who had 8 children: a. William Bourke b. Walter c. Henry d. Joseph e. Arthur f. Duke g. Mary h. Josephine
If this is your family, I can give you prior generations and succeeding ones on different branches, one of whom went to America, and another who is my own ancestor.