Polariton pulses-transient light-matter hybrid excitations-traveling through anisotropic media can lead to unusual optical phenomena in space and time. However, studying these pulses presents challenges with their anisotropic, ultrafast, and nanoscale field variations. Here, we demonstrate the creation, observation, and control of polariton pulses, with in-plane hyperbolic dispersion, on anisotropic crystal surfaces by using a time-resolved nanoimaging technique and our developed high-dimensional data processing. We capture and analyze movies of distinctive pulse spatiotemporal dynamics, including curved ultraslow energy flow trajectories, anisotropic dissipation, and dynamical misalignment between phase and group velocities. Our approach enables analysis of polariton pulses in the wave vector time domain, demonstrating a time-domain polaritonic topological transition from lenticular to hyperbolic dispersion contours and the ability to study the polariton-induced time-varying optical forces. Our findings promise to facilitate the study of diverse space-time phenomena at extreme scales and drive advances in ultrafast nanoimaging.