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We consider here point processes $N^f(t)$, $t>0$, with independent incrementsand integer-valued jumps whose distribution is expressed in terms ofBern\v{s}tein functions $f$ with L\'evy measure $\nu$. We obtain the generalexpression of the probability generating functions $G^f$ of $N^f$, theequations governing the state probabilities $p_k^f$ of $N^f$, and theircorresponding explicit forms. We also give the distribution of thefirst-passage times $T_k^f$ of $N^f$, and the related governing equation. Westudy in detail the cases of the fractional Poisson process, the relativisticPoisson process and the Gamma Poisson process whose state probabilities havethe form of a negative binomial. The distribution of the times $\tau_j^{l_j}$of jumps with height $l_j$ ($\sum_{j=1}^rl_j = k$) under the condition $N(t) =k$ for all these special processes is investigated in detail.

Counting processes with Bern\v{s}tein intertimes and random jumps