  We employ a time-dependent real-space local density-of-states method to study
the movement and fusion of Majorana zero modes in the 1D interacting Kitaev
model, based on the time evolution of many-body states.
  We analyze the dynamics and both fusion channels of Majoranas using
time-dependent potentials, either creating {\it Walls} or {\it Wells}. %
focusing on the local density-of-states and charge-density of fermions varying
with time. For fast moving Majoranas, we unveil non-equilibrium signatures of
the ``strong-zero mode'' operator (quasi parity degeneracy in the full
spectrum) and its breakdown in the presence of repulsive Coulomb interactions.
Focusing on forming a full electron after fusion, we also discuss upper and
lower limits on the Majorana speed needed to reduce non-adiabatic effects and
to avoid poisoning due to decoherence.
