Understanding how individual edges influence the behavior of graph neural networks (GNNs) is essential for improving their interpretability and robustness. Graph influence functions have emerged as promising tools to efficiently estimate the effects of edge deletions without retraining. However, existing influence prediction methods rely on strict convexity assumptions, exclusively consider the influence of edge deletions while disregarding edge insertions, and fail to capture changes in message propagation caused by these modifications. In this work, we propose a proximal Bregman response function specifically tailored for GNNs, relaxing the convexity requirement and enabling accurate influence prediction for standard neural network architectures. Furthermore, our method explicitly accounts for message propagation effects and extends influence prediction to both edge deletions and insertions in a principled way. Experiments with real-world datasets demonstrate accurate influence predictions for different characteristics of GNNs. We further demonstrate that the influence function is versatile in applications such as graph rewiring and adversarial attacks.