  This paper studies the first hitting times of generalized Poisson processes
$N^f(t)$, related to Bernstein functions $f$. For the space-fractional Poisson
processes, $N^\alpha(t)$, $t>0$ (corresponding to $f= x^\alpha$), the hitting
probabilities $P\{T_k^\alpha<\infty\}$ are explicitly obtained and analyzed.
The processes $N^f(t)$ are time-changed Poisson processes $N(H^f(t))$ with
subordinators $H^f(t)$ and here we study $N\left(\sum_{j=1}^n
H^{f_j}(t)\right)$ and obtain probabilistic features of these extended counting
processes. A section of the paper is devoted to processes of the form
$N(|\mathcal{G}_{H,\nu}(t)|)$ where $\mathcal{G}_{H,\nu}(t)$ are generalized
grey Brownian motions. This involves the theory of time-dependent fractional
operators of the McBride form. While the time-fractional Poisson process is a
renewal process, we prove that the space-time Poisson process is no longer a
renewal process.
