Using a previously described projection technique for magnetic resonance images (MRI), we have studied the dynamics of kinematic waves arising from frequency gradients in an anisotropically cooling, oscillating Belousov-Zhabotinsky (BZ) medium. The gradual conversion of purely kinematic waves into diffusion-driven, chemical waves is illustrated by comparing the time dependence of the experimental wave velocity with the velocity calculated from a model for an idealized kinematic wave that excludes diffusive effects. Finally, we have studied the interaction of kinematic waves and trigger (chemical) waves. These measurements provide the explanation for previous observations that had suggested sudden changes of wavefront velocities.