Building on ideas introduced by Eshelby in 1953, and on recent dynamical extensions of the Peierls model for screw and edge dislocations, an approximate equation of motion (EoM) to govern non-uniform dislocation motion under time-varying stress is derived, allowing for time variations of the core width. Non-local in time, it accounts for radiative visco-inertial effects and non-radiative drag. It is completely determined by energy functions computed at constant velocity. Various limits are examined, including that of vanishing core width. Known results are retrieved as particular cases. Notably, the EoM reduces to Rosakis's Model I for steady motion [Rosakis, P., 2001. Supersonic dislocation kinetics from an augmented Peierls model. Phys. Rev. Lett. 86, 95-98]. The frequency-dependent effective response coefficients are obtained within the linearized theory, and the dynamical self-force is studied for abrupt or smooth velocity changes accompanied by core variations in the full theory. A quantitative distinction is made between low- and high-acceleration regimes, in relation to occurrence of time-logarithmic behavior.