  The present paper is the natural continuation of the letter [Phys. Rev. Lett.
\textbf{113}, 040402 (2014)], where the electron wave functions in the presence
of a background electromagnetic field of general space-time structure have been
constructed analytically, assuming that the initial energy of the electron is
the largest dynamical energy scale in the problem and having in mind the case
of a background tightly focused laser beam. Here, we determine the scalar and
the spinor propagators under the same approximations, which are useful tools
for calculating, e.g., total probabilities of processes occurring in such
complex electromagnetic fields. In addition, we also present a simpler and more
general expression of the electron wave functions found in [Phys. Rev. Lett.
\textbf{113}, 040402 (2014)] and we indicate a substitution rule to obtain them
starting from the well-known Volkov wave functions in a plane-wave field.
