When food coloring is added to water, it spreads throughout the water in a process called diffusion. The coloring continues to spread until all parts of the water contain an equal concentration of the dye.
Diffusion of the food coloring occurs passively, meaning no outside force is required for it to happen. However, the process can be sped up by heating, shaking or stirring the water. Heating the water excites the dye particles, causing them to move more quickly. Stirring or shaking physically moves the food coloring throughout the solution.
Once an even concentration is reached, the dye molecules continue to move, but the movement is no longer evident to the human eye.