A variational theory is developed for the study of collisional transport in a multi-species plasma immersed in an equilibrium magnetic field defined in a general non-symmetric and bounded domain. The present approach allows, in principle, the consideration of the exact Fokker-Planck collision operator appearing in the kinetic equation for the one-particle distribution function of each species and the determination of the transport coefficients relevant for a macroscopic description of the dynamical system.