There are several branches in the current family treebut the underlying current of the training is much the same. A form of Japanese Jujutsu (soft method) where sogo bujutsu ( total martial training ) philosophy is applied.
They believe in fighting in many ranges both armed and un-armed. Emphasis is using an opponents tactics and strengths against them. The idea is to yield to oncomming force and then reverse the momentum of the situation back into the opponent ( think of some one pushing on a tree branch and letting go so it smacks them in the face).
The original Yoshin-ryu jujutsu as founded by Akiyama Shirobei Yoshitoki no longer exists and probably died out sometime in the late 19th Century. The only remnant of the system is the Naginata system which is still extant in Hiroshima Prefecture, Japan.
Hontai Yoshin-ryu has nothing to do with Akiyama's Yoshin-ryu. HYR actually stems from the line of systems founded by Takagi Oriemon Shigetoshi and Takagi Umanosuke shigesada (Takagi-ryu, Hontai Takagi Yoshin-ryu, Takagi Yoshin-ryu etc, etc.)
The systems still extant that are related to Akiyama's Yoshin-ryu are Tenjin Shinyo-ryu jujutsu (An amalgamation of Yoshin-ryu and Shin no shindo-ryu) and Shinto Yoshin-ryu jujutsu. Tenjin Shinyo-ryu at present, is only taught in Japan, Australia and the UK.