  A family of Feller branching diffusions $Z^x$, $x \ge 0$, with nonlinear
drift and initial value $x$ can, with a suitable coupling over the {\em
ancestral masses} $x$, be viewed as a path-valued process indexed by $x$. For a
coupling due to Dawson and Li, which in case of a linear drift describes the
corresponding Feller branching diffusion, and in our case makes the path-valued
process Markovian, we find an SDE solved by $Z$, which is driven by a random
point measure on excursion space. In this way we are able to identify the
infinitesimal generator of the path-valued process. We also establish path
properties of $x\mapsto Z^x$ using various couplings of $Z$ with classical
Feller branching diffusions.
