How can a rainbow be so steady, even though the droplets causing it can be in such different states? The rainbow! What magical "thing". And even if you see the droplets of rain move in a sunlit storm, she's steady. I have been trying to understand but there are so many drops involved! And they are moving in turbulent ways on top.
So what's going on? I know that each droplet sends a "rain circle" cone towards us, and that our eyes are sprayed with these cones. These cones all have the same orientation, no matter how the droplets move. Somehow this must be the key, but I don't see how.
 A: The color depends on the relative angle between you, the drop, and the sun.
If you were to track an individual drop it would change color as it falls "through" the rainbow. That would be cool to see!
In more detail:
Consider a cartoon similar to the one in @John Hunter's answer. There is a viewer at left looking right. There is a sun behind the viewer. For every point to the right of the viewer we can draw lines from that point to both the viewer and the sun. So we correspond an angle to EVERY point in space.
We can ask "what are the surfaces of constant angle?" Well, imagine a line through the sun and the viewer. All points along this line will be angle zero. Larger angles will be cones coaxial with this center line. These are the surfaces of constant angle.
A rainbow works because, for certain angles, the droplet will reflect the various spectral components of the sun to the viewer.
So what is fixed in space, given the position of the viewer and sun, are the cones of constant angle. These are present whether or not there is water. A rainbow arises when there are water droplets occupying the appropriate cones.
As a droplet falls, it passes through the different cones, meaning the component of the sun that it reflects changes in time as it passes through the cones of differing angles.
A: This picture was posted on this answer, Rainbows and Clouds might be handy again now

The drops are in continuous motion but the two shown will quickly be replaced by others in the same positions...
(although the image has been regularly used on Stack exchange: Copyright 1999 Rebecca McDowell rebeccapaton.net/rainbows/formatn.htm )
A: The apparent colour of each particle depends on the position, but only on “2D position in view”, i.e. angular position relative to the observer. (See other answers and questions for explanation on this.) Thus it doesn't matter how chaotic they move: whereever each particle is, it will have the “correct” colour to fit in the rainbow.

