I maybe able to help a little bit.
It appears the family lived in Vryoula (Asia Minor) or around Vryoula. Off course, during 1921 and 1922 many Greeks left Asia Minor and did seek refuge else where. It just happens that I have a list of all refugees that migrated to Crete.
The 43th meeting in Heraklion, Crete, dated 17 Dec 1922 accepted one Kantaras family as refugees.
Kantaras Demitrios (son of Kyriakos) age 45 from Vryoula
Lygeria (wife of the above) age 40
Irene (daughter of the above) age 15
Stelios (son of the above) age 12
Maria (daughter of the above) age 8
Ekaterini (daughter of the above) age 3
Stamatia (daughter of the above) age 1
Fortunately, the names above are in the second page. Unfortunately, the listing of the names is very complex and the family names are not in alphabetic order. There were over 60,000 refugees, that makes searching difficult,