The solar corona is continually being injected with magnetic flux from the solar interior, flux that cannot be annihilated in bulk by the electrically highly conducting corona. Observations have shown that mass loss through downflows within solar prominences may be responsible for the ascent and eventual eruption of these prominences. Ascending small-scale structures as well as large-scale eruptions of prominences may both serve to carry excess magnetic flux out of the corona. We investigate the dissipation and field reconnection across the tangential discontinuities that form spontaneously in the supporting magnetic field of a prominence. Our analysis of a variety of postdissipation, nonequilibrium states provides instructive insights into the observable motions in prominences. The net effect of the magnetic reconnection is a downflow of mass accompanied by an upward transport of magnetic flux. This effect may play an important, direct and indirect, role in ejecting magnetic flux from the corona into interplanetary space. Shock waves are a natural consequence of this dissipative process, and their detection may serve as a diagnostic observational tool. Thin counterstreaming layers of prominence plasma predicted by this work have already been observed.