The millimeter wave (mmWave) frequencies offer the potential for enormous capacity wireless links. However, designing robust communication systems at these frequencies requires that we understand the channel dynamics over both time and space: mmWave signals are extremely vulnerable to blocking and the channel can thus rapidly appear and disappear with small movement of obstacles and reflectors. In rich scattering environments, different paths may experience different blocking trajectories and understanding these multi-path blocking dynamics is essential for developing and assessing beamforming and beam-tracking algorithms. This paper presents the design and experimental results of a novel measurement system which uses phased arrays to perform mmWave dynamic channel measurements. Specifically, human blockage and its effects across multiple paths are investigated with only several microseconds between successive measurements. From these measurements we develop a modeling technique which uses low-rank tensor factorization to separate the available paths so that their joint statistics can be understood.