It sounds like you can animate the water falling, then duplicate the bones and click
Yes to also duplicate the keys. Now you have two sets of bones, each with all the images and both sets are animating the same. Now you need to offset all the keys for one of the bones.
Normally offsetting is done for a looping animation, but the water falling fades out at the end, it doesn't loop. The offset tool requires the same keys at the start and end. You could copy keys from frame 0 to the frame after the water fades out, then you can use the offset tool and delete the extra keys afterward.
Otherwise you can offset manually. To do that, the keys for one set of bones:
Go to the last frame and ctrl+shift+L to key everything that is visible:
Copy the new keys you just created and paste them on frame 0:
Now move all your keys after the original last frame of the animation to the beginning, so the last keys are just before the original start keys:
You can set those last keys to stepped, so there's no transition between the last key and the first key. You can also hold
shift and move them very close to the original start keys.
Note I didn't have keys between the cut point and the last frame, but you'd move all keys after the original last frame to the beginning.
All that is roughly the same as what the offset tool is doing for you. You move your keys past the end of the animation, then you cut the animation and move the keys that are past the end back to the beginning.
Here's the ugly animation I made when making the above screenshots:
It shows a bone and mesh whose keys were offset.