In spatial transcriptomics (ST) data, biologically relevant features such as tissue compartments or cell-state transitions are reflected by gene expression gradients. Here, we present SpaceWalker, a visual analytics tool for exploring the local gradient structure of 2D and 3D ST data. The user can be guided by the local intrinsic dimensionality of the high-dimensional data to define seed locations, from which a flood-fill algorithm identifies transcriptomically similar cells on the fly, based on the high-dimensional data topology. In several use cases, we demonstrate that the spatial projection of these flooded cells highlights tissue architectural features and that interactive retrieval of gene expression gradients in the spatial and transcriptomic domains confirms known biology. We also show that SpaceWalker generalizes to several different ST protocols and scales well to large, multi-slice, 3D whole-brain ST data while maintaining real-time interaction performance.