We consider the problem of in-network compressed sensing, where the goal is to recover a global, sparse signal from local measurements using only local computation and communication. Our approach to this distributed compressed sensing problem is based on the centralized Iterative Hard Thresholding algorithm (IHT). In time-varying networks, the network dynamics necessarily introduce inaccuracies that are not present in a centralized implementation of IHT. To accommodate these inaccuracies, we show how centralized IHT can be extended to include inexact computations while still providing the same recovery guarantees. We then leverage these new theoretical results to develop a distributed version of IHT for dynamic networks. Evaluations show that our algorithm outperforms the best-known existing solution in both time and bandwidth by several orders of magnitude.