I agree with both Hatte's hypothesis.
There is an interesting French Family Tree which also lists the father and son as Moshe, but qualifies that records before 1500 are not quite reliable, though the Horowitz family seems to be a bit of an exception.
ie.
Family Tree of Famille Quevy-Dumont:
http://gw.geneanet.org/fquevy?lang=en;pz=elise+marie+paule+jeanne;n...
Father : http://gw.geneanet.org/fquevy?lang=en;pz=elise+marie+paule+jeanne;n...
Son: http://gw.geneanet.org/fquevy?lang=en;pz=elise+marie+paule+jeanne;n...
Born in 1488 - Mainz , 55001, Mainz, Rheinland Pfalz, Allemagne
". . . However, I want to clarify that certain data contained beyond 1500 are mostly historical or sometimes for some hypothetical and therefore can always be questioned. The Jewish ancestry of the Horowitz family has been studied for centuries and is relatively reliable."
Note that the family name Ashkenazi is ironically enough classified as of Iberian origins.
https://www.familytreedna.com/public/IberianSurnamesofAshkenaz/
(Schelley Dardashti)
Following the Ashkenazi surname ancestry back, it is indeed not Sephardi, but are direct descendants of Rashi from Treves, France.
http://www.loebtree.com/rashi.html
Daniel Loeb mentions, 117 [44] Rabbi (TREVESIAN) Jochanan (Yahanan I) Ashkenazi TREVES (b. Abt 1265 Treves, d. Worms, Germany). Left France between 1285 and 1314 during the expulsion of Jews by Philippe IV and settled in Germany where he acquired the surname Ashkenazi.
Private User BTW, Vivian, I am a 15th great granddaughter of Moshe Ashkenazi too.
We are connected via Nechama Ashkenazi (Teomim) who was a Frankel Teomim.
According to Geni you and I are 13th cousins!
You are my 11th cousin. Common ancestor is http://www.geni.com/people/Yom-Tov-Lippman-Heller-תוספות-יו-ט/60000...