The difference between electron and muon neutrino charged-current cross sections has attracted quite some interest over the past few years. This interest is guided by the experimental effort that aims at measuring the CP-violating phase by looking for electron (anti-)neutrino appearance in muon neutrino beams. In long-baseline experiments such as T2K, models for the neutrino cross section are often constrained by near-detector data, with a muon neutrino flux that is unoscillated. Non-trivial differences between electron and muon neutrino cross sections are currently experimentally not well constrained, and different models give varying results, especially in kinematic regions where nuclear structure details become important, i.e. for low energy and momentum transfers. In this work we present the nuclear response and cross section using different nuclear models, for forward lepton scattering in the region of a couple 100 MeVs. In this kinematic region the cross section is sensitive to nuclear structure details which are not accounted for in simplified models such as the relativistic Fermi gas (RFG) which is commonly used in the experimental analysis. The results show that it is important for current and future accelerator-based experiments, notably T2K and the short-baseline oscillation program (i.e the MicroBooNE, SBND and ICARUS experiments) which are sensitive to the several 100 MeV region, to take nuclear structure details into account in their analysis.