The dynamic behavior of synthetic gene circuits plays a key role in ensuring their correct function. However, our ability to accurately predict this dynamic behavior is limited by our quantitative understanding of the circuits being constructed. This understanding can be represented as a mathematical model, which can be used to optimize circuit performance and inform the design of future circuits. Previous inference methods have used fluorescent reporters to quantify average behaviors over an extended time window, resulting in a static characterization which is a poor predictor of dynamics. Here we present a method for characterizing the dynamic behavior of synthetic gene circuits. The method relies on parameter inference techniques applied to time-series measurements of cell cultures growing in microtiter plates. We use our method to design and characterize gene circuits in E. coli that provide core functionality for engineering cell behavior at the population level. We arrange 23 biological parts into 9 devices and combine them to construct and measure 9 gene circuits including relays, receivers and a degrader. We demonstrate that the behaviors of simple devices can be modeled dynamically and used to predict the behaviors of more complex circuits. Furthermore, our method allows incremental inference of models as new circuits are constructed, and lays the foundation for iteratively learning dynamic models from data in a scalable manner.