Dye sublimation can be used on any surface where substrate polymer treatments can work, from mugs to glass to snowboarding.
Since the polymer is the basis of the dyestuff sublimation, if this substrate can maintain the polymer treatment on its surface, it can be sealed with the dyestuff. When heat and pressure are applied to the processed item and bonded to the printed heat transfer paper, the polymer expands and opens, like a flower in the sun.
While printing the custom sublimated shirts heat under pressure causes the polymer to expand, heat causes the dye to gas (about 400ºF) and the dye (as a gas) to flow into the exposed polymer, creating a nice continuous hue on cooling that enhances color durability.
The polymer spray will pick up can be screen-printed. Much has been written about sublimation printing on cotton t-shirts, but actual sublimation printing cannot be done on natural fabrics, although there are some heat transfer inks that work as a sublimation process.
The problem with natural fibers is that they don't open and close like polymer fibers. Because they don't open or close but are porous as the dye becomes a gas and seeps through the fiber, sticking to the surface rather than the fiber. If you wash the fabric again, the color will return.
Cotton has been successfully printed for decades, but the problem has always been with the feel of the ink, which fades over time with successive washes, leaving behind a slightly strong hand feeling. However, there are some new inks mentioned above that are soft to the touch and as close to sublimation without polyester as possible.