Question

How does water move on its own concentration gradient?

Answer 1

I'm pretty sure it cant move on its own

Answer 2

Are you speaking of a density gradient, in which a more concentrated solution moves below a less concentrated solution?

In that case, the more concentrated solution has the greater density, and it will gradually sink below the less concentrated solution. 

In the same way, a stone will sink in water, which is less dense than the stone.