In this paper, different types of Poisson processes N subordinated to random time processes X, depending on Brownian motion, are analysed. In particular, the processes X considered here are the elastic Brownian motion , the Brownian sojourn time on the positive half-line the first-passage time (through the level t) of a Brownian motion, with or without drift, and the -Bessel process , for . In all these cases, we obtain the explicit state probability distributions , , , their governing difference-differential equations and some moments. The connections among different models and, in particular, of with birth and death processes are obtained and discussed. The models presented have probability distributions governed by higher order or fractional difference-differential equations. In the case of the Poisson process with Bessel times, equations with time-dependent coefficients appear.