To extend our previous Green's-function approach for steady-state hot-electron transport to the transient region, coupled differential and integral equations are constructed for the evolution of the drift velocity v (t) and the electron temperature Te(t). The effect of current fluctuations on these equations is included and discussed qualitatively. When the evolution equation for v(t) is linear-ized, a Langevin equation is obtained for the drift velocity. For electric fields with moderate strengths, we show that the calculated values for v(t) as a function of t with memory and without memory differ very little. Thus the memory effect is negligible and the original evolution equations reduce to a pair of differential and nonlinear equations for v(t) and Te(t). Numerical computation has been carried out for hole transport in p-type Ge. Scatterings due to both acoustic and nonpolar-optical phonons are considered. For a step field at t--0, v(t) shows the well-known ballis-tic and overshoot behavior. When a rectangular pulse field is applied, both overshoot and un-dershoot phenomena may appear in v (t).