Fine-scale characterization and monitoring of spatiotemporal processes are crucial for high-performance quality control of manufacturing processes, such as ultrasonic metal welding and high-precision machining. However, it is generally expensive to acquire high-resolution spatiotemporal data in manufacturing due to the high cost of the 3D measurement system or the time-consuming measurement process. In this paper, we develop a novel dynamic sampling design algorithm to cost-effectively characterize spatiotemporal processes in manufacturing. A spatiotemporal state-space model and Kalman filter are used to predictively determine the measurement locations using a criterion considering both the prediction performance and the measurement cost. The determination of measurement locations is formulated as a binary integer programming problem, and genetic algorithm is applied for searching the optimal design. In addition, a new test statistic is proposed to monitor and update the surface progression rate. Both simulated and real-world spatiotemporal data are used to demonstrate the effectiveness of the proposed method.