  We discuss the QED-generalized leading-twist light-cone distribution
amplitudes of heavy mesons, that appear in QCD$\times$QED factorization
theorems for exclusive two-body $B$ decays. In the presence of electrically
charged particles, these functions should be more appropriately regarded as
soft functions for heavy-meson decays into two back-to-back particles. In this
paper, we derive the one-loop anomalous dimension of these soft functions and
study their behaviour under renormalization-scale evolution, obtaining an exact
solution in Laplace space. In addition, we provide numerical solutions for the
soft functions and analytical solutions to all orders in the strong and to
first order in the electromagnetic coupling. For the inverse (and
inverse-logarithmic) moments, we obtain an all-order solution in both
couplings. We further provide numerical estimates for QED corrections to the
inverse moments.
