Dear Stefano, You know well that family history in Middle East is generally an oral one, you couldn't show your Greek or Crusader origin during and under the Memluke yoke! but if we look to history we can get some points:
1- Douai is the capital of the ancient Frankish Duchy of Ostrevant ( the arms of the Douai family are used as a blason of Ostrevant till now), in the First Crusade,Anselme d'Ostrevant was the "right hand" of Godfroy de Bouillon in the East, so one would assume that the progenitors were related to the D'Ostrevant family... and if Jeremiah Al-Amshiti had been one of the Douai clan, the family progenitors would have been "easternized" just after the First Crusade.
2- Philippe de Tarraze, a great Syro-Lebanese historian, always said that many of the Douaihy clan came from Sadad in the Desert between Homs and Palmyra in Syria, he counted some branches of the Douaihy there: Karkour, Tadros, Wehbi, Shehadeh, Obeid... so we can assume that after the Crusades the Douaihy clan distributed in the Christian regions in Syria and Lebanon, between Ehden in Lebanon, and Aleppo and Sadad in Syria, they are still many branches in Syria, the mentioned above in Sadad, and the Ayyoub and Hawwa in Aleppo, besides the branches in Northern Palestine ( from the Hawwa clan which is originated in Douaihy), and the "Kubba" family, in Aleppo and Livorno!
Do you know whether the Tadros in Northern Lebanon are related to the Douaihy, some told me that they are from the Frangieh family and others say that they are from Prince family in Tripoli, while Philippe de Tarraze counted them as a branch of the Douaihi ( at least in Syrian Sadad).
If you want, send me your email, I have the arms of the "de Douai" and "d'ostrevant" families, that are my estimation concerning the history of the Douaihy, I am interested in those families who are descendants of the Crusders in Syria and Lebanon. Regards