They do both of those things. They can also lose color and remain on the plant, either because of disease issue or genetics.

With fall color hormones are triggered by daylight length and this causes the abscission layer to form which blocks the transport of chlorophyll (green) so xanthophylls (yellow) and carotenoids (orange) are expressed and trapped sugars in the leaf form anthocyanins (red/purple). As the abscission layer dries it causes the leaf to separate. It would seem that the leaf would lose the ability to produce food for the plant as soon as the chlorophyll is blocked. I don't know if the process is the same if it's due to disease. BTW I don't remember this from botany class, too long ago, I just looked it up. lol