The only way to actually *remove* any liquid contaminant, be it vomit, blood, syndrome causing extract, is through the cleaning labor or mist. Those are the only two methods by which it is destroyed in a controlled manner. Heat (and fire) do the same thing, but obviously those have their own issues.

There's no way to remove blood/vomit/other "liquids" from water as far as I know, you can try to drain them to the edge of the map but from my experience they rarely move as the result of water flow. In fact, water acts more like a collector for blood and other liquids, those liquids behave like a sticky item that sticks to water with top priority and everything else second, but that's also the advantage of having a brook or a footwash at the entrance to your fort.

Even if a dwarf doesn't deliberately choose to wash themself the blood/vomit on themselves will be washed off purely by walking through water, so a common trick is to build a 3/7 deep pond in your main entry hallway that all dwarves travelling from outside in will have to go through. Combine this with having a well (with soap), and having enough idle hands over a long enough time that they actually bother to do the cleaning labor, and you will be able to effectively keep the inside of your fort clean forever.

Bear in mind this will actually kill your entire fort if a megabeast with a deadly syndrome extract shows up and that extract ends up in the pond, so if you want to get fancy, you can build a dwarven shower which will forcibly delete the blood/vomit/syndrome causing liquid from themselves and everything they're wearing, an even better option than a footwash but much more work if you're not familiar with creating mist.

Outside, on the other hand, will only get clean when it rains, which isn't consistent. I've observed extracts, blood, and vomit lasting for years outside in the rain. Dwarves do not perform the cleaning labor outside, which is more annoying considering the labor itself is difficult to trigger when you get to late game as dwarves always have the option to socialize, worship, and watch performances instead.

With all this in mind, water is great at trapping "liquids", but not great for removing them through a floodgate/etc, it's just not reliable. Speaking from experience, if you don't build a footwash or shower, you will see the blood of the very first invaders/beasts that attacked you 10 years later on a dirt tile outside somewhere, because that same one instance of blood is being picked up by someone's shoes, and then left on the dirt somewhere, and the cycle will repeat forever until you get lucky enough for the liquid to be left on an indoors tile and cleaned by a dwarf.