The time evolution of a plasma confined in a tandem mirror with thermal barriers has been studied. A physics model is given which describes the kinetic interactions in velocity space between the particles of the various plasma species that exist in each spatial region of the confinement, and the effects of a variety of particle and energy sources applied to the plasma. The analysis includes particle and energy rate equations for the various species determining the plasma confinement. The analysis also includes quasi-neutrality and ambipolarity conditions which define the ambipolar potential profile along the axis of the device as well as expressions for the passing particle densities in each region. This model describes in a self-consistent manner the time evolution of tandem mirror confinement with thermal barriers including the steady-state phase of operation. The resulting system of equations is solved numerically. The axicell MFTF-B configuration has been studied specifically. A possible startup scenario has been obtained. The results show the time sequence that must be followed to build up a plasma from given initial conditions to the steady-state phase by means of appropriately programmed particle and energy sources applied to the plasma.