Although it is generally accepted that structural parameters like width, shape, and edge structure crucially affect the electronic characteristics of graphene nanoribbons (GNRs), the exact relationship between geometry and charge transport remains largely unexplored. In this paper, we present in situ through-transport measurements of various topological GNRs and GNR heterostructures by lifting the ribbon with the tip of a scanning tunneling microscope. At the same time, we develop a comprehensive transport model that enables us to understand various features, such as obscuring of localized states in through transport, the effect of topology on transport, as well as negative differential conductance in heterostructures with localized electronic modes. The combined experimental and theoretical efforts described in this paper serve to elucidate general charge transport phenomena in GNRs and GNR heterostructures.