Interacting systems with K driven particle species on an open chain or chains that are coupled at the ends to boundary reservoirs with fixed particle densities are considered. We classify discontinuous and continuous phase transitions that are driven by adiabatic change of boundary conditions. We build minimal paths along which any given boundary-driven phase transition (BDPT) is observed and we reveal kinetic mechanisms governing these transitions. Combining minimal paths, we can drive the system from a stationary state with all positive characteristic speeds to a state with all negative characteristic speeds, by means of adiabatic changes of the boundary conditions. We show that along such composite paths, one generically encounters Z discontinuous and 2(K-Z) continuous BDPT's, with Z taking values 0≥Z≥K depending on the path. As model examples, we consider solvable exclusion processes with product measure states and K=1,2,3 particle species and a nonsolvable two-way traffic model. Our findings are confirmed by numerical integration of hydrodynamic limit equations and by Monte Carlo simulations. Results extend straightforwardly to a wide class of driven diffusive systems with several conserved particle species.