GPUs are flexible parallel processors capable of accelerating real applications. To exploit them, programmers must ensure a consistent program state between the CPU and GPU memories by managing data. Manually managing data is tedious and error-prone. In prior work on automatic CPU-GPU data management, alias analysis quality limits performance, and type-inference quality limits applicability. This paper presents Dynamically Managed Data (DyManD), the first automatic system to manage complex and recursive data-structures without static analyses. By replacing static analyses with a dynamic run-time system, DyManD overcomes the performance limitations of alias analysis and enables management for complex and recursive data-structures. DyManD-enabled GPU parallelization matches the performance of prior work equipped with perfectly precise alias analysis for 27 programs and demonstrates improved applicability on programs not previously managed automatically.