Although deep learning-based methods have been successfully applied to polarimetric synthetic aperture radar (PolSAR) image classification tasks, most of the available techniques are not suitable to deal with PolSAR data on irregular domains, e.g., superpixel graphs, because they are naturally designed as grid-based architectures in Euclidean space. To overcome this limitation and achieve robust PolSAR image classification, this article proposes the multiscale evolving weighted graph convolutional network, where weighted graphs based on superpixel technique and Wishart-derived distance are constructed to enable efficient handling of graphical PolSAR data representations. In this article, we derive a new architectural design named graph evolving module that combines pairwise latent feature similarity and kernel diffusion to refine the graph structure in each scale. Finally, we propose a graph integration module based on self-attention to perform robust hierarchical feature extraction and learn an optimal linear combination of various scales to exploit effective feature propagation on multiple graphs. We validate the superiority of proposed approach on classification performance with four real-measured datasets and demonstrate significant improvements compared to state-of-the-art methods. Additionally, the proposed method has shown strong generalization capacity across datasets with similar land covers.