The ability to detect motion and to track a moving object that is hidden around a corner or behind a wall provides a crucial advantage when physically going around the obstacle is impossible or dangerous. One recently demonstrated approach to achieving this goal makes use of non-line-of-sight picosecond pulse laser ranging. This approach has recently become interesting due to the availability of single-photon avalanche diode (SPAD) receivers with picosecond time resolution. We present a time-resolved non-sequential ray-tracing model and its application to indirect line-of-sight detection of moving targets. The model makes use of the Zemax optical design programme's capabilities in stray light analysis where it traces large numbers of rays through multiple random scattering events in a 3D non-sequential environment. Our model then reconstructs the generated multi-segment ray paths and adds temporal analysis. Validation of this model against experimental results is shown. We then exercise the model to explore the limits placed on system design by available laser sources and detectors. In particular we detail the requirements on the laser's pulse energy, duration and repetition rate, and on the receiver's temporal response and sensitivity. These are discussed in terms of the resulting implications for achievable range, resolution and measurement time while retaining eye-safety with this technique. Finally, the model is used to examine potential extensions to the experimental system that may allow for increased localisation of the position of the detected moving object, such as the inclusion of multiple detectors and/or multiple emitters.