Reversible logic has recently progressed in superconducting circuits and promises momentous improvements in computing efficiency relative to irreversible logic developed by industry in semiconductor circuits. Superconducting logic uses single flux quanta (SFQ) to represent bits, and unlike their irreversible SFQ-logic relatives the bits may switch state with extremely low dissipation. Here we propose and simulate ballistic shift registers (BSRs). These devices are reversible and dramatically more energy-efficient than previous ones, including a pioneering one developed before the advent of SFQ logic. The BSR is ballistic because no power is supplied to the gate other than the input bits themselves; the results are conveyed as slightly slower output bits. The BSR uses an SFQ of either polarity for the stored bit as well as ballistic fluxons. The BSR additionally uses shunt capacitors and the nonlinearity of the ends of long Josephson junctions to support partial fluxons and energy-conserving dynamics. The BSRs are multi-port and allow asynchronous bit arrivalimportant functionalities not available in previous reversible gates. During their interaction, the stored and the moving bits swap their bit states, resulting in the shift-register operation. We show that two BSRs operate in sequence without power added. In addition, we show that a BSR circuit with two inputs allows a bit state to be conveyed on different outputs. This constitutes the first asynchronous ballistic 2-input gate. Finally, for a more intuitive understanding of the dynamics, we introduce a collective coordinate model. It describes the 1-bit BSR dynamics with the motion of two coordinates in a potential determined by the states of the input bit and the initial stored SFQ. We discuss the potential for this SFQ logic type in the context of energy efficiency, parameter margins, logical depth, and speed.