We present new evidence that the bright nonthermal X-ray emission features in the interior of the Cassiopeia A supernova remnant are caused by inward-moving shocks, based on Chandra and NuSTAR observations. 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 transverse velocities were estimated to be in the range of similar to 2100-3800 km s(-1) for a distance of 3.4 kpc. The shock velocities in the frame of the expanding ejecta reach values of similar to 5100-8700 km s(-1), which is slightly higher than the typical speed of the forward shock. Additionally, we find flux variations (both increasing and decreasing) on timescales of a few years in some of the inward-moving shock filaments. The rapid variability timescales are consistent with an amplified magnetic field of B similar to 0.5-1 mG. The high speed and low photon cut-off energy of the inward-moving shocks are shown to imply a particle diffusion coefficient that departs from the Bohm regime (k(0) = D-0/D-0,D-Bohm similar to 3-8) for the few simple physical configurations we consider in this study. The maximum electron energy at these shocks is estimated to be similar to 8-11 TeV, which is smaller than the values of similar to 15-34 TeV that were inferred for 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.