We examine the privacy amplification of channels that do not necessarily satisfy any LDP guarantee by analyzing their contraction behavior in terms of $f_α$-divergence, an $f$-divergence related to Rényi-divergence via a monotonic transformation. We present bounds on contraction for restricted sets of prior distributions via $f$-divergence inequalities and present an improved Pinsker's inequality for $f_α$-divergence based on the joint range technique by Harremoës and Vajda. The presented bound is tight whenever the value of the total variation distance is larger than 1/$α$. By applying these inequalities in a cross-channel setting, we arrive at strong data processing inequalities for $f_α$-divergence that can be adapted to use-case specific restrictions of input distributions and channel. The application of these results to privacy amplification shows that even very sparse channels can lead to significant privacy amplification when used as a post-processing step after local differentially private mechanisms.