​This paper studies the first hitting times of generalized Poisson processes Nf(t), related to Bernstein functions f. For the space-fractional Poisson processes, Nα(t), t>0 (corresponding to f=xα), the hitting probabilities P{Tαk<∞}are explicitly obtained and analyzed. The processes Nf(t) are time-changed Poisson processes N(Hf(t))with subordinators Hf(t) and here we study N(∑nj=1Hfj(t))and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form N(|GH,ν(t)|)where GH,ν(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.