We investigate the mixing times of Markov kernels under $E_\gamma$-divergence. We demonstrate that the zero-error $E_\gamma$-mixing time, for any $\gamma>1$, of irreducible and aperiodic Markov chains, is bounded, a property that is not shared by the TV-mixing time. We further obtain upper bounds on the $E_\gamma$-mixing times for a broad family of contractive Markov kernels via a new non-linear strong data processing inequality for the $E_\gamma$-divergence. We apply our results to derive new bounds for the local differential privacy guarantees offered by the sequential application of a privacy mechanism to data.