The invention provides novel high-energy density and low-cost flow electrochemical devices incorporating solid-flow electrodes, and further provides methods of using such electrochemical devices. Included are anode and cathode current collector foils that can be made to move during discharge or recharge of the device. Solid-flow devices according to the invention provide improved charging capability due to direct replacement of the conventional electrode stack, higher volumetric and gravimetric energy density, and reduced battery cost due to reduced dimensions of the ion-permeable layer.