In statistical analysis of video sequences for speech recognition, and more generally activity recognition, it is natural to treat temporal evolutions of features as trajectories on Riemannian manifolds. However, different evolution patterns result in arbitrary parameterizations of these trajectories. We investigate a recent framework from statistics literature that handles this nuisance variability using a cost function/distance for temporal registration and statistical summarization & modeling of trajectories. It is based on a mathematical representation of trajectories, termed transported square-root vector field (TSRVF), and the L2 norm on the space of TSRVFs. We apply this framework to the problem of speech recognition using both audio and visual components. In each case, we extract features, form trajectories on corresponding manifolds, and compute parametrization-invariant distances using TSRVFs for speech classification. On the OuluVS database the classification performance under metric increases significantly, by nearly 100% under both modalities and for all choices of features. We obtained speaker-dependent classification rate of 70% and 96% for visual and audio components, respectively.