Image relighting modifies illumination while preserving non-lighting content such as identity and geometry. Existing diffusion-based methods often suffer from unstable illumination changes or inconsistent content preservation under complex lighting, as they lack an explicit mechanism to learn feature transformations between images. We reformulate relighting as an illumination feature transport problem and introduce Consistent Feature Transport (CFT), a training principle that explicitly enforces illumination-consistent transport between source and target image distributions. Built upon rectified flow, CFT jointly models noise-to-image generation and illumination-consistent source-to-target transport through trajectory-level supervision. This dual-transport formulation encourages isolation of illumination-specific variations while preserving content-aligned features. To support complex lighting scenarios, we construct a large-scale portrait relighting dataset with diverse relighting effects. Experiments show consistent improvements over existing state-of-the-art relighting approaches and demonstrate that CFT can generalize to other editing tasks, including style transfer. Code is available at https://github.com/Dixin-Lab/CFT.