The kinematics of supernova remnants (SNRs) provide useful information on asymmetries in the distribution of the ambient medium and in the explosions themselves.  In particular, for young SNRs with ages up to several hundred years, we can investigate motions of the shock-heated ejecta in detail using X-ray monitoring observations. Here we present new results on the motions of X-ray features in two young SNRs: Kepler’s SNR and Cassiopeia A, using Chandra observations.  Through a combination of both spectroscopy and imaging with Chandra, we measured radial (Doppler) velocities and transverse (proper motion) velocities of several small features in Kepler’s SNR.  We found an aspherical expansion with a velocity range ~1,000-10,000 km/s for Kepler. The asymmetries in Kepler appear to be caused by a highly asymmetric distribution of ambient material, likely due to mass loss from the companion star, which would tend to support a single-degenerate origin. We also revised the location of the explosion center using our kinematic information. Our new accurate explosion site location has well-defined positional uncertainties, allowing for a great reduction in the area to be searched for faint surviving donor stars under non-traditional single-degenerate SNe Ia scenarios; because of the lack of bright stars in the search area the traditional scenario remains ruled out.  We will also show the physical properties of mysterious “inward-moving shocks” in Cassiopeia A. Several bright inward-moving filaments were identified using monitoring data taken by Chandra in 2000-2014. These inward-moving shock locations are nearly coincident with hard X-ray (15-40 keV) hot spots seen by NuSTAR. From proper-motion measurements, the shock velocities in the frame of the expanding ejecta reach values of ~‪5100-8700‬ km/s, which is slightly higher than the typical speed of the forward shock. Cassiopeia A is dynamically too young for its reverse shock to appear to be moving inward in the observer frame. We propose instead that the inward-moving shocks are a consequence of the forward shock encountering a density jump of < ~5-8 in the surrounding material. Thus the interaction with ambient dense material (e.g., a molecular cloud) might be the reason for the strong particle acceleration we see at the reverse shock in Cassiopeia A.