We introduce and study branching interval partition diffusions in their natural generality. We let interval widths evolve independently according to a general real-valued diffusion subject only to conditions that ensure finite lifetimes of intervals and allow the continuous generation of new intervals. The latter is governed by the Pitman-Yor excursion measure of the real-valued diffusion and an associated spectrally positive Lévy process to order both these excursions and their start times. This generalises previous work by Forman, Pal, Rizzolo and Winkel on the self-similar case and gives rise to a new class of general Markovian homogeneous Crump-Mode-Jagers-type branching processes with characteristics varying diffusively during their lifetimes.