We study the dynamics of a directed manifold of internal dimension D in a d-dimensional quenched random force field. We obtain an exact solution for d → ∞ and a Hartree approximation for finite d. They yield a Flory-like roughness exponent ζ and a nontrivial anomalous diffusion exponent ν continuously dependent on the ratio gT/gL of divergence-free (gT) to potential (gL) disorder strength. For the particle (D = 0) our results agree with previous order epsilon2 RG calculations. The time-translational invariant dynamics for gT > 0 smoothly crosses over to the previously studied ultrametric aging solution in the potential case.