What happens to carbon during diffusion?

By definition, it diffuses. In some metals, Fe is the prime example, C is soluble in solid metal. If you pack a piece of pure Fe in a source of C and heat it to the right temperature for an appropriate time, the Fe becomes "carburized". The C atoms diffuse into the Fe and fit into interstitial sites in the Fe lattice. These C atoms strengthen the Fe and make it harder. If you continue this long enough the C content increases to the point where the Fe and C form Fe3C, iron carbide. Control of C content in Fe produces steel. Today there are thousands of different steel alloys with various levels of C and other alloying elements.