Heat transfer in essence is the transport of thermal energy along certain paths in a similar way as fluid motion is the transport of fluid parcels along fluid paths. This similarity admits Lagrangian heat-transfer analyses by the geometry of such "thermal paths" analogous to well-known Lagrangian mixing analyses. Essential to Lagrangian heat-transfer formalisms is the reference state for the convective flux. Existing approaches admit only uniform references. However, for convective heat transfer, a case of great practical relevance, the conductive state that sets in for vanishing fluid motion is the more natural reference. This typically is an inhomogeneous state and thus beyond the existing formalism. The present study closes this gap by its generalisation to non-uniform references and thus substantially strengthens Lagrangian methods for thermal analyses. This ansatz is demonstrated by way of a 2D case study and offers new fundamental insight into thermal transport that is complementary to the Eulerian picture based on temperature and heat-transfer coefficients