The only source I have yet to find acknowledging this switch in clothing from a Sephardic perspective is in the English edition to the Yalkut Yosef Hilkhot Shabbat. Under Siman 242, Halakhah 5, regarding the mitzvah to change from weekday clothes into more elegant garments, the editor (R. Yisrael Bitan) added a special footnote:
The Kabbalists ruled that one must wear white suits on Shabbat. Even
so, in our time all the greatest Torah scholars are accustomed to wear
black suits on Shabbat, and one should not act differently. If an
individual chooses to be different from everyone else and wear a white
suit, he is acting improperly; he is compared to a groom sitting among
mourners. It important to explain this to those who begin observing
mitzvot on their own (ba'ale teshuvah) so that they will conduct
themselves as recommended by the Torah leaders of our generation.
This is directly opposing the Ben Ish Chai's position not to wear any black on shabbat (Ben Ish Chai, 2nd year, Lech Lecha #18). I also just came across an article from R. Marc Angel on the subject. He mentions personally talking with R. Mordecai Eliyahu, in the early 90's, on the Ashkenazic-assimilated dress code:
Rabbi Eliyahu responded: the Ashkenazic garb has become the "standard" garb for Talmidei Hakhamim, and Sephardic rabbis won't be taken seriously enough if they don't dress according to this fashion. When I said that the situation might be turned around if he and other Sephardic leaders made an issue of it, he said it wasn't worth it and it wouldn't succeed.
I haven't found a source explaining the process of assimilating the clothing, but I do remember hearing a shiur from R. Rakeffet, who explained that R. Ovadia's children went to Ashkenazic yeshivot because those institutions had the highest standard of Torah learning at the time. Such an environment, bearing in mind the already existing struggles of a stereotyped minority, could likely create the norm R. Eliyahu noted.