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

Continuous time random walk and diffusion with generalized fractional Poisson process