We propose a time-varying graph signal recovery method for estimating the true time-varying graph signal from corrupted observations by leveraging dynamic graphs. Most of the conventional methods for time-varying graph signal recovery have been proposed under the assumption that the underlying graph that houses the signals is static. However, in light of rapid advances in sensor technology, the assumption that sensor networks are time-varying like the signals is becoming a very practical problem setting. In this paper, we focus on such cases and formulate dynamic graph signal recovery as a constrained convex optimization problem that simultaneously estimates both time-varying graph signals and sparsely modeled outliers. In our formulation, we use two types of regularizations, time-varying graph Laplacian-based and temporal differencebased, and also separately modeled missing values with known positions and unknown outliers to achieve robust estimations from highly degraded data. In addition, an algorithm is developed to efficiently solve the optimization problem based on a primaldual splitting method. Extensive experiments on simulated drone remote sensing data and real-world sea surface temperature data demonstrate the advantages of the proposed method over existing methods.