Excess emission over expected diffuse astrophysical backgrounds in the direction of the Galactic center region has been claimed at various wavelengths, from radio to gamma rays. Among particle models advocated to explain such observations, a few invoke interactions between dark matter particles and ordinary matter, such as cosmic rays, interstellar gas or free electrons. Depending on the specific interstellar matter particles' species and energy, such models predict distinct morphological features. In this study we make detailed, model-independent predictions for the morphology of models where the relevant electromagnetic emission is proportional to the product of the dark matter density profile and the density of interstellar matter or cosmic rays. We compare the predicted latitudinal and longitudinal distributions with observations, and provide the associated set of relevant spatial templates.