[en] The space-time propagator for multichannel scattering is cast in the form of a Feynman path integral where the action is a matrix and the interactions are nonlocal in time. The S matrix is evaluated in the semiclassical limit, leading to trajectories which naturally follow individual channel potentials in noninteracting regions and adiabatic-like ones in strongly interacting ones. A numerical example is provided for the curve-crossing problem, and the agreement with exact quantum results is good even when the interaction region is large and/or encompasses classical turning points