His daughter is Avigail Buso [originally Abuchatzeira but then she married a person by the name of David Buso and became Mrs. Buso]. She had a lot of insights about her father. And her children also. One of her children Moshe Buso receives visitors in Jerusalem.
However it does seem that there is a careful guarding of the name Abuchatzeira in such a way that there is great jealously about who gets the name and who gets to be considered as passing on the mantle of greatness.
It is well known that there was a ten year period of rivalry between R. Baruch in Netivot and R. Elazar in Beer Sheva.
First of all, it should be known that although something miraculous was going on in that family for about five generations, it does seemed to have come to a halt. There is no member of the Abuchatzeira family today that you can go to and be guaranteed to come back with a miracle. However some drop of that holiness and special quality does remain in the Buso Family.
The most basic thing that made Bava Sali special was his personal service towards God. Without that kind of self sacrifice no one can claim the Bava Sali mantle. One does not become a tzadik/saint by reason of birth, but by choice.
In general the path of Bava Sali was what is described by classical Musar books like the Chovot Levavaot [Duties of the Heart]. Its world view is essentially that of Conservative Judaism --Mesorati Judaism.
Today if someone would want a blessing in their life, my recommendation would to go to Shimon Buso in Netivot. He is the closest thing that I can see to the continuance of the Bava Sali Tradition and Path.
[ that Shimon Buso does not officially receive visitors. Still I think one who wants a blessing would be wise to go to him.]
Bava Sali, did learn Kabalah. But that was not in any way the thing that stood out by him. His thing was just plain, old, simple service of God. (Just the old, old time religion was his thing.) And service of God was a personal thing for him. It did not include any of the fanaticism of the general Ashkenazi world.
The general approach is that as much of divine service that one can do--all the better. But everyone was accepted by Bava Sali. No one ever came away without a clear and definite miracle. The only exception was when once he visited Jerusalem. And some kabalist wanted to visit him, and he refused to see him. In fact, all kabalists were rigorously eliminated from his list of visitors, according to Bava Sali's daughter, Mrs. Buso.
Mrs Buso also once summed up for me the essence of Bava Sali's path-. But this was understood very differently that the fanatic Ashkenazim understand Shulchan Aruch. For example, Mrs. Buso regularly listened to classical music--including specifically the Hallelujah Chorus of Handel. Moshe Buso asked me on several occasions to play for him the violin concerto of Mendelssohn.
I did spend some time learning the Eitz Chaim of the Ari with Shimon Buso. I was with that family in around 1991 and 1992 and then later I reconnected in around 2000-2003. Then there was a third period around 2009-2010. So I do not know what is going on there today. But from the entire family of Bava Sali and his older brother David Abuchatzeira and their grandfather Rav Yaakov I can say their path was a simple as possible--straight Litvak Torah: Talmud and Musar of Reb Israel Salanter.
The mot interesting thing about that path is not what it does but what it excludes. Any all all frills and extras, all kinds of frumkeit that people want to add were totally missing.