We present a general solution to the collisionless Boltzmann (Vlasov) equation for a free-flowing plasma along a magnetic field line using Liouville's theorem, allowing for an arbitrary field-aligned potential energy structure including nonmonotonicities. The constraints of the existing collisionless kinetic transport models are explored, and the need for a more general approach to the problem of self-consistent potential energy calculations is described. Then a technique that handles an arbitrary potential energy distribution along the field line is presented and discussed. For precipitation of magnetospherically trapped hot plasma, this model yields moment calculations that vary by up to a factor of 2 for various potential energy structures with the same total potential energy drop. The differences are much greater for the high-latitude outflow scenario, giving order of magnitude variations depending on the shape of the potential energy distribution. Self-consistent calculations for the photoelectron-driven polar wind are compared with previous results, and it is shown that even a photoelectron concentration of 0.03% at the base of the simulation (500 km) will cause the potential energy distribution to violate the constraints of the existing models. ¿ 1998 American Geophysical Union