We report a systematic experimental and theoretical investigation of core-shell InGaN/GaN single wire light-emitting diodes (LEDs) using electron beam induced current (EBIC) microscopy. The wires were grown by catalyst-free MOVPE and processed into single wire LEDs using electron beam lithography on dispersed wires. The influence of the acceleration voltage and of the applied bias on the EBIC maps was investigated. We show that the EBIC maps provide information both on the minority carrier effects (i.e. on the local p-n junction collection efficiency) and on the majority carrier effects (i.e. the transport efficiency from the excited region toward the contacts). Because of a finite core and shell resistance a non-negligible current redistribution into the p-n junction takes place during the majority carrier transport. A theoretical model for transport in a core-shell wire is developed, allowing to explain the dependence of the EBIC profiles on the experimental parameters (the electron beam acceleration voltage and the bias applied on the device) and on the structural parameters of the wire (core and shell resistance, shunt resistance, etc). Comparison between simulated and experimental profiles provides valuable information concerning the structure inhomogeneities and gives insight into the wire electrical parameters.