Carbon migrates in hot steel. At forging temps, it migrates quite well. The surface is exposed to oxygen and the surface carbon becomes carbon dioxide, and thus leaves the steel. The result is a small carbon loss. This loss is fairly shallow, but with repeated heating cycles, a significant carbon loss can be affected, especially with higher temperatures. That is why forging cycles should be kept to as few as are needed.
As to adding carbon, it can be increased by exposing the steel to an atmosphere of low oxygen, and high carbon. This can be done in a special atmosphere oven, or by coating the surface with a chemical that will readily release carbon ( or nitrogen ,or boron) into the steel surface. This is a diffusion process, and the amount of diffusion is quite shallow. The total addition of carbon to the blade is very small, and only on the surface. If the blade was ground and sanded, all the additional carbon would be lost.
To my knowledge, there is no way to directly "Pound" carbon into a blade by general forging of a blade.
The techniques of adding carbon to low carbon steel and wrought iron were by adding a "flux" of carbon rich ash ( rice straw ash in the case of tamahagane, and coal dust in blister and plow steels) and folding the steel many times. The continued folding and incorporation of the carbon, plus its migration caused the steel to have a higher carbon content. The distribution was not homogeneous, however. The boundaries of the welds and the silicon slag create layers of higher and lower carbon content. This is the source of the hada in well forged tamahagane, and the pattern in wootz and similar steels ( and the fibers in wrought iron).
