  A non-Markovian counting process, the `generalized fractional Poisson
process' (GFPP) introduced by Cahoy and Polito in 2013 is analyzed. The GFPP
contains two index parameters $0<\beta\leq 1$, $\alpha >0$ and a time scale
parameter. Generalizations to Laskin's fractional Poisson distribution and to
the fractional Kolmogorov-Feller equation are derived. We develop a continuous
time random walk subordinated to a GFPP in the infinite integer lattice
$\mathbb{Z}^d$. For this stochastic motion, we deduce a `generalized fractional
diffusion equation'. In a well-scaled diffusion limit this motion is governed
by the same type of fractional diffusion equation as with the fractional
Poisson process exhibiting subdiffusive $t^{\beta}$-power law for the
mean-square displacement. In the special cases $\alpha=1$ with $0<\beta<1$ the
equations of the Laskin fractional Poisson process and for $\alpha=1$ with
$\beta=1$ the classical equations of the standard Poisson process are
recovered. The remarkably rich dynamics introduced by the GFPP opens a wide
field of applications in anomalous transport and in the dynamics of complex
systems.
