Use white vinegar to remove food coloring from skin, and for tough stains, use a baking soda and water paste. Be sure to rinse the vinegar from the skin first if switching to the paste.
Use warm water when rinsing, and if the stain is still persistent, alternate between the vinegar rub and the baking soda paste. If children get the food coloring on them, try not to get either mixture in their mouth. Additionally, whether on an adult or child, try not to scrub the skin to the point of damage. The food coloring will come out, so it is not necessary to scrub harshly.