Recent advances in mobile robotic platforms like quadruped robots and drones have spurred a demand for deploying visuomotor policies in increasingly dynamic environments. However, the collection of high-quality training data, the impact of platform motion and processing delays, and limited onboard computing resources pose significant barriers to existing solutions. In this work, we present STDArm, a system that directly transfers policies trained under static conditions to dynamic platforms without extensive modifications. The core of STDArm is a real-time action correction framework consisting of: (1) an action manager to boost control frequency and maintain temporal consistency, (2) a stabilizer with a lightweight prediction network to compensate for motion disturbances, and (3) an online latency estimation module for calibrating system parameters. In this way, STDArm achieves centimeter-level precision in mobile manipulation tasks. We conduct comprehensive evaluations of the proposed STDArm on two types of robotic arms, four types of mobile platforms, and three tasks. Experimental results indicate that the STDArm enables real-time compensation for platform motion disturbances while preserving the original policy's manipulation capabilities, achieving centimeter-level operational precision during robot motion.