Channel-specific transit time distributions for a two-level model system exhibiting simple avoided crossing are computed and interpreted in terms of a general semiclassical analysis which expresses wave packet evolution in terms of sums over complex trajectories. Novel effects observed include "quantum speedup" (the Hartman effect), for tunneling, and "quantum slowdown" for above-barrier reflection. Nonadiabatic effects are found to appear only at times for which their exponentially small amplitude is comparable or greater than the amplitude of adiabatic contributions. This is in accord with the semiclassical treatment provided. Nonadiabatic effects include, most notably, the appearance of slowly leaking resonances in the excited level potential energy well.