I’ve heard that there are many Ashkenazim who fill teaching posts at Sephardic Yeshivot. I am sure that they are very worthy teachers, but whether they know the Ben Ish Chai backwards and forwards like they do the Kitzur Shulchan Aruch is another matter. Some Sephardim must see leaving the training of the next generation of pulpit rabbis in the hands of Ashkenazim, however well intentioned, as a threat to their continuity and cultural traditions.
I would also point out that post-Shoah yeshivot have gone back to the sources, and have de-emphasized learning the minhagim. After so many years of this, we are hardly aware that it was not always so. Inevitably, the products of Ashkenazic yeshivot would teach as they were taught, whether that is at an Ashkenazic or Sephardic Yeshiva.