Charged-particle transport is characterized by scattering cross sections that are extremely large and forward-peaked, requiring specialized treatment as compared with neutral-particle transport. The extended-transport correction (ETC) is known to be an effective method to treat elastic scattering of electrons. We apply the ETC to inelastic downscattering of electrons, and evaluate the effectiveness of the method by comparing the scattering moments for the screened Rutherford scattering kernel and for scattering with a deterministic cosine. The ETC approximation results in a -function in angle downscatter source term, for energy loss without direction change, which has been incorporated into the CEPTRE discrete ordinates code in a manner that is compatible with general quadrature sets, not requiring a specialized Galerkin quadrature. The ETC approximation also makes it possible to develop a first-collision source technique that is effective for charged-particle transport, by including particles that have downscattered in energy without direction change in the uncollided-flux solution. We demonstrate the effectiveness of these techniques for problems involving electron beam sources incident on infinite and finite water cylinders and compare the energy- and charge-deposition distributions with ITS Monte Carlo results with good agreement.