Which classical radiation-reaction equation is the classical limit of QED

The Lorentz-Abraham-Dirac (LAD) equation, third order in time with characteristic time $\tau_0 = 2 e^2/(3 m c^3) \sim 6.3e-24\,\mathrm{s}$, has runaway and preaccelerating solutions; restricting LAD to its non-runaway (critical) manifold gives a second-order equation whose first-order approximation in $\tau_0$ is the Landau-Lifshitz (LL) equation; the two differ from order $\tau_0^2$ on. Determine the classical limit ($\hbar \to 0$ at fixed classical field parameters) of QED for a point charge in a background field to all orders in $\tau_0$, and identify whether it coincides with LL, with LAD on the critical manifold, or with another second-order equation (for example Ford-O'Connell), at orders beyond the leading one in $\alpha$.

A derivation from QED in a general (at least plane-wave) background of the classical equation of motion to second and higher order in $\tau_{0}$, compared term by term with LL and with the critical-manifold reduction of LAD.