Agents learn about a state using private signals and the past actions of their neighbors in a network. In contrast to most such models, the target being learned about is moving around. We ask: when can a group aggregate information quickly, keeping up with the changing environment? First, if private signals are diverse enough in their precisions, then Bayesian learning achieves good information aggregation as long as individuals observe sufficiently many others. Second, without diversity in signal distributions, Bayesian information aggregation can fall far short of good aggregation benchmarks, and even be Pareto-inefficient. Third, good aggregation requires anti-imitation; without it, agents’ estimates are inefficiently confounded by “echoes” of past perceptions. At a technical level, stationary equilibria of Bayesian learning are characterized by linear rules reminiscent of the simple DeGroot heuristic, with coefficients satisfying a certain system of equations. The resulting tractability can facilitate structural estimation of equilibrium learning models and testing against behavioral alternatives, as well as the analysis of welfare and influence.