We revisit Approximate Graph Propagation (AGP), a unified framework which captures various graph propagation tasks, such as PageRank, feature propagation in Graph Neural Networks (GNNs), and graph-based Retrieval-Augmented Generation (RAG). Our work focuses on the settings of dynamic graphs and dynamic parameterized queries, where the underlying graphs evolve over time (updated by edge insertions or deletions) and the input query parameters are specified on the fly to fit application needs. Our first contribution is an interesting observation that the SOTA solution, AGP-Static, can be adapted to support dynamic parameterized queries; however several challenges remain unresolved. Firstly, the query time complexity of AGP-Static is based on an assumption of using an optimal algorithm for subset sampling in its query algorithm. Unfortunately, back to that time, such an algorithm did not exist; without such an optimal algorithm, an extra $O(\log^2 n)$ factor is required in the query complexity, where $n$ is the number of vertices in the graphs. Secondly, AGP-Static performs poorly on dynamic graphs, taking $O(n\log n)$ time to process each update. To address these challenges, we propose a new algorithm, AGP-Static++, which is simpler yet reduces roughly a factor of $O(\log^2 n)$ in the query complexity while preserving the approximation guarantees of AGP-Static. However, AGP-Static++ still requires $O(n)$ time to process each update. To better support dynamic graphs, we further propose AGP-Dynamic, which achieves $O(1)$ amortized time per update, significantly improving the aforementioned $O(n)$ per-update bound, while still preserving the query complexity and approximation guarantees. Last, our comprehensive experiments validate the theoretical improvements: compared to the baselines, our algorithm achieves speedups of up to $177\times$ on update time and $10\times$ on query efficiency.