We discuss reaction-diffusion systems with reactions satisfying mass-action kinetics and the detailed-balance condition. They allow for a gradient structure where the driving functional is the relative entropy and the dissipation potential is the sum of a Wasserstein part for diffusion and a reaction part. This formulation highlights the additive splitting of the dissipative processes for diffusion in terms of optimal transport and for reaction in terms of transformation in chemical space. We emphasize the geometric structure induced by the interaction of these two dissipative processes. For the simple case of a scalar reaction-diffusion equation the induced dissipation distance can be characterized explicitly, giving the so-called Hellinger-Kantorovich distance on the set of all measure-valued concentration profiles.