Our partial-current-transport (PCT) approach uses the partial currents through the faces of cells in a spatial grid as the unknowns in a linear algebra problem. Emission and externally incident currents are the knowns. The coefficient matrix is determined by boundary conditions and transport within cells. Adaptive PCT models include within-cell flux-distribution parameters that are found by distribution iteration (DI). Upon convergence, scalar fluxes are computed. We develop the approach in general and derive (in slab geometry) a fixed-coefficient PCT diffusion method and an adaptive PCT discrete ordinates method. A parallelized direct solver is used for the large but very sparse linear algebra problem that couples all the cells. Matrix inversion is used for the dense but small within-cell problems. These direct solvers eliminate scattering source iteration (SI). Though requiring more storage, much or most of the computational effort is pleasingly parallel, making the method attractive for large parallel machines with large memories. In comparing our slab geometry implementation with PARTISN, we observed that DI used as many or fewer iterations than SI and succeeded where SI failed, whether alone or with diffusion synthetic acceleration or transport synthetic acceleration. We conclude that DI for adaptive PCT holds great promise as an alternative to SI and its accelerators.