  First-principles studies of strongly-interacting hadronic systems using
lattice quantum chromodynamics (QCD) have been complemented in recent years
with the inclusion of quantum electrodynamics (QED). The aim is to confront
experimental results with more precise theoretical determinations, e.g. for the
anomalous magnetic moment of the muon and the CP-violating parameters in the
decay of mesons. Quantifying the effects arising from enclosing QED in a finite
volume remains a primary target of investigations. To this end, finite-volume
corrections to hadron masses in the presence of QED have been carefully studied
in recent years. This paper extends such studies to the self-energy of moving
charged hadrons, both on and away from their mass shell. In particular, we
present analytical results for leading finite-volume corrections to the
self-energy of spin-0 and spin-$\frac{1}{2}$ particles in the presence of QED
on a periodic hypercubic lattice, once the spatial zero mode of the photon is
removed, a framework that is called $\mathrm{QED}_{\mathrm{L}}$. By altering
modes beyond the zero mode, an improvement scheme is introduced to eliminate
the leading finite-volume corrections to masses, with potential applications to
other hadronic quantities. Our analytical results are verified by a dedicated
numerical study of a lattice scalar field theory coupled to
$\mathrm{QED}_{\mathrm{L}}$. Further, this paper offers new perspectives on the
subtleties involved in applying low-energy effective field theories in the
presence of $\mathrm{QED}_{\mathrm{L}}$, a theory that is rendered non-local
with the exclusion of the spatial zero mode of the photon, clarifying recent
discussions on this matter.
