The equations governing the diffusion and transport of a fully collisional plasma across a strong magnetic field in a bounded domain are analyzed. Following a relatively short relaxation time, the diffusion exhibits universal properties independent of the choice of initial data. Mathematically this appears as a time asymptotic solution which is space-time separable. The temporal decay rate is a nonlinear eigenvalue which is found via the solution of a related eigenvalue problem. This determines the spatial distribution of both the particle density and the pressure. Some of the transport effects caused by Bremsstrahlung radiation, particle, and heat injection are considered, and the conditions under which the system evolves into an equilibrium are examined.