We characterize dynamic (not just steady state) equilibria in a search-theoretic model of fiat money, where buyers and sellers, upon meeting, enter bargaining games to determine prices.  Equilibrium in the bargaining game is approximated in terms of a tractable dynamical system, in much the same way that the Nash solution approximates equilibrium in bargaining games in stationary environments.  The model with our dynamic bargaining solution can generate outcomes (such as limit cycles) that never arise in the same model if one imposes a myopic bargaining solution, as has been done in the past.