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.
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