  We construct a semiclassical theory for the transport of topological
junctions starting from a microscopic Hamiltonian that comprehensively includes
the interplay among the Majorana qubit, the Josephson phase, and the
dissipation process. With the path integral approach, we derive a set of
semiclassical equations of motion that can be used to calculate the time
evolution of the Josephson phase and the Majorana qubit. In the equations we
reveal rich dynamical phenomena such as the qubit induced charge pumping, the
effective spin-orbit torque, and the Gilbert damping. We demonstrate the
influence of these dynamical phenomena on the transport signatures of the
junction. We apply the theory to study the Shapiro steps of the junction, and
find the suppression of the first Shapiro step due to the dynamical feedback of
the Majorana qubit.
