We introduce a class of processing architectures for node features on a graph, that are equivariant with respect to local actions of a general symmetry group G, i.e. G may act on each feature \(\varphi _{v}\) at some node v by an individual transformation \(g_{v}\in G\) . Our method is based on modelling node feature data in terms of sections \(\varGamma (E)\) of associated vector bundles, which ensures preservation of local symmetries by construction. Processing architectures are then derived from mappings of bundle sections \(\mathcal {F} : \varGamma (E) \rightarrow \varGamma (E)\) . We focus on mappings \(\mathcal {F}\) induced by diffusion PDEs on vector bundles associated to a generalized Laplacian. We define a bundle scale space, which contains non-trivial fixed points corresponding to harmonic sections of the associated vector bundle, in contrast to the basic Gaussian scale space. We utilize vector diffusion maps and lattice gauge theory in order to discretize the geometric PDEs such that local symmetries are still respected. We outline parametrizations suited for supervised machine learning scenarios and provide a proof-of-concept numerical experiment.