The colour of feed that a cow eats has absolutely nothing to do with the colour of the milk. Milk is always going to be white no matter what a cow eats. The quality of what that cow eats, however, does play a far greater role in milk production. This is why dairy cows need to be fed high-quality feeds in order to produce as much milk, and has high quality of milk, as they are able.
The process of milk production is not directly tied to a cow's digestive system. It's actually tied by the cow's circulatory system that carries nutrients taken from the digestive tract to the mammary glands.
Basically, then, milk is made from the cow's blood. The only way milk can be made from blood is by tiny alveoli found in the upper reaches of the udder. The cells that make up each alveoli are supplied by capillaries, which are the tiniest blood vessels in the body. These cells (called "milk cells") are responsible for synthesizing and secreting milk from blood.
Milk has similar components to that of blood, minus the hemoglobins and red blood cells that gives blood its red colour. All these components—lipids, amino acids (which collectively form larger protein structures), vitamins, minerals, water, and other components and compounds that enter the milk cells—are secreted into the alveoli in the form they came in, or as a modified component to add to what would now be called “milk.” Casein is a major contributor that makes milk white, along with the lipids.
Milk goes from the alveoli into ducts, which lead to larger ducts and so on and so forth, until it ends up in the gland cistern of that quarter (since a cow’s udder is divided into four quarters). The milk in the gland cistern goes on into the teat cistern or sinus, where it sits until the suction from a calf’s mouth, the hands of a human (who knows how to properly hand-milk a cow), or the gentle vacuum suction of the milking machine's suction cups stimulate the sphincter muscles at the base of the teat to relax and release the milk. Oxytocin is a hormone that is also responsible for encouraging "milk let-down," which is released from the cow's brain as, basically, a lactating stimulus.
Naturally, there is far more complexities and science than what was explained here, but this is basically what happens when milk is produced, and how a cow's milk (or any mammal's milk for that matter) is white when she eats green forage.
The Bovine Practicum Q&A "blog" is an informative, just-for-fun section where I find a variety of questions that are often (and not-so-often) asked by inquisitive people like you, and answer them to the best of my ability. Much of the questions have come from a wide variety of reaches from the Internet.
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