ABSTRACT We present a per cent-level accurate model of the line-of-sight velocity distribution of galaxies around dark matter haloes as a function of projected radius and halo mass. The model is developed and tested using synthetic galaxy catalogues generated with the UniverseMachine run on the Multi-Dark Planck 2 N-body simulations. The model decomposes the galaxies around a cluster into three kinematically distinct classes: orbiting, infalling, and interloping galaxies. We demonstrate that: (1) we can statistically distinguish between these three types of galaxies using only projected line-of-sight velocity information; (2) the halo edge radius inferred from the line-of-sight velocity dispersion is an excellent proxy for the three-dimensional halo edge radius; and (3) we can accurately recover the full velocity dispersion profile for each of the three populations of galaxies. Importantly, the velocity dispersion profiles of the orbiting and infalling galaxies contain five independent parameters – three distinct radial scales and two velocity dispersion amplitudes – each of which is correlated with mass. Thus, the velocity dispersion profile of galaxy clusters has inherent redundancies that allow us to perform non-trivial systematics checks from a single data set. We discuss several potential applications of our new model for detecting the edge radius and constraining cosmology and astrophysics using upcoming spectroscopic surveys.