We investigate the concept of Augmented Dynamic Data Physicalization, the combination of shape-changing physical data representations with high-resolution virtual content. Tangible data sculptures, for example using mid-air shape-changing interfaces, are aesthetically appealing and persistent, but also technically and spatially limited. Blending them with Augmented Reality overlays such as scales, labels, or other contextual information opens up new possibilities. We explore the potential of this promising combination and propose a set of essential visualization components and interaction principles. They facilitate sophisticated hybrid data visualizations, for example Overview & Detail techniques or 3D view aggregations. We discuss three implemented applications that demonstrate how our approach can be used for personal information hubs, interactive exhibitions, and immersive data analytics. Based on these use cases, we conducted hands-on sessions with external experts, resulting in valuable feedback and insights. They highlight the potential of combining dynamic physicalizations with dynamic AR overlays to create rich and engaging data experiences.