  We consider two fractional versions of a family of nonnegative integer valued
processes. We prove that their probability mass functions solve fractional
Kolmogorov forward equations, and we show the overdispersion of these
processes. As particular examples in this family, we can define fractional
versions of some processes in the literature as the Polya-Aeppli, the Poisson
Inverse Gaussian and the Negative Binomial. We also define and study some more
general fractional versions with two fractional parameters.
