Flow matching-based generative models have been integrated into the plug-and-play image restoration framework, and the resulting plug-and-play flow matching (PnP-Flow) model has achieved some remarkable empirical success for image restoration. However, the theoretical understanding of PnP-Flow lags its empirical success. In this paper, we derive a continuous limit for PnP-Flow, resulting in a stochastic differential equation (SDE) surrogate model of PnP-Flow. The SDE model provides two particular insights to improve PnP-Flow for image restoration: (1) It enables us to quantify the error for image restoration, informing us to improve step scheduling and regularize the Lipschitz constant of the neural network-parameterized vector field for error reduction. (2) It informs us to accelerate off-the-shelf PnP-Flow models via extrapolation, resulting in a rescaled version of the proposed SDE model. We validate the efficacy of the SDE-informed improved PnP-Flow using several benchmark tasks, including image denoising, deblurring, super-resolution, and inpainting. Numerical results show that our method significantly outperforms the baseline PnP-Flow and other state-of-the-art approaches, achieving superior performance across evaluation metrics.