We present new evidence that the bright non-thermal X-ray emission features in the interior of the Cassiopeia A supernova remnant (SNR) 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 $sim$2,100-3,800 km s$^{-1}$ for a distance of 3.4 kpc. The shock velocities in the frame of the expanding ejecta reach values of $sim$5,100-8,700 km s$^{-1}$, 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 $B sim$ 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 ($k_0 = D_0/D_{rm 0,Bohm} sim$ 3-8) for the few simple physical configurations we consider in this study. The maximum electron energy at these shocks is estimated to be $sim$8-11 TeV, smaller than the values of $sim$15-34 TeV 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 $gtrsim$ 5-8 in the surrounding material.
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