The mesmerising movement of ink through water is in fact nothing more than the random motion of molecules.
Ink enters the fluid as one dark blob, before diffusing from this original region of high concentration to regions in the water with lower concentration.
This happens because randomly moving water molecules collide with ink molecules, causing them to spin off in different directions. As regions with more ink molecules undergo more collisions with water molecules, the ink molecules from these regions will, on average, move into areas with fewer molecules.
The distance these ink molecules travel through water depends on their size; bigger molecules are slowed down by friction more than smaller molecules.