We study dynamical correlations for the Aldous-Diaconis-Hammersley process, a one-dimensional totally asymmetric interacting particle system.  Its stationary measures, indexed by particle density, are Poisson point processes on the line.  We relate the stationary two-point function to the transition probability of a second class particle.  The space-time trajectories of particles can be mapped to contour lines of a last passage percolation problem with boundary sources, studied by Baik and Rains [JSP 100(3):523(2000)].  Using their result we express the rescaled correlation function in terms of the Riemann-Hilbert problem from Painlevé II.  Finally we discuss other initial conditions like shocks and deterministic with constant spacing.