  We derive contour integral formulas for the real space propagator of the
spin-$\tfrac12$ XXZ chain. The exact results are valid in any finite volume
with periodic boundary conditions, and for any value of the anisotropy
parameter. The integrals are on fixed contours, that are independent of the
Bethe Ansatz solution of the model and the string hypothesis. The propagator is
obtained by two different methods. First we compute it through the spectral sum
of a deformed model, and as a by-product we also compute the propagator of the
XXZ chain perturbed by a Dzyaloshinskii-Moriya interaction term. As a second
way we also compute the propagator through a lattice path integral, which is
evaluated exactly utilizing the so-called $F$-basis in the mirror (or quantum)
channel. The final expressions are similar to the Yudson representation of the
infinite volume propagator, with the volume entering as a parameter. As an
application of the propagator we compute the Loschmidt amplitude for the
quantum quench from a domain wall state.
