In this paper, a framework for dynamic high-dimensional hypothesis testing in wireless sensor networks is presented. The sensor nodes (SNs) collect and transmit to a fusion center (FC), in a distributed fashion, compressed measurements of a time-correlated hypothesis vector. The FC, based on the measurements collected, tracks the hypothesis vector, and feeds back minimal information about the uncertainty in the current estimate, which enables adaptation of the SNs' data collection and transmission strategy. The policy of the SNs is optimized with the overall objective of minimizing the detection error probability, under sensing and transmission cost constraints incurred by each SN. A Bernoulli approximation on the detection error is employed, which enables a significant reduction in the optimization complexity and the design of scalable estimators based on sparse approximation recovery algorithms. Simulation results demonstrate that, for a target 5% detection error, the adaptive scheme attains 90% and 50% cost savings with respect to a memoryless scheme which does not exploit the time-correlation and a non-adaptive one, respectively.