We study the problem of describing local components of height functions on abelian varieties over characteristic 0 local fields as functions on spaces of torsors under various realisations of a 2-step unipotent motivic fundamental group naturally associated to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the Q_l and Q_p-pro-unipotent etale realisations when the base field is p-adic, and in terms of the R-pro-unipotent Betti-de Rham realisation when the base field is archimedean. In the course of proving the p-adic instance of these theorems, we develop a new technique for studying local non-abelian Bloch-Kato Selmer sets, working with certain explicit cosimplicial group models for these sets and using methods from homotopical algebra. Among other uses, these models enable us to construct a non-abelian generalisation of the Bloch-Kato exponential sequence under minimal conditions. On the geometric side, we also prove a number of foundational results on local constancy or analyticity of various non-abelian Kummer maps for pro-unipotent etale or Betti-de Rham fundamental groups of arbitrary smooth varieties.