In classical configuration spaces, you can take a single point in the configuration space, and the single point describes the entire state of a classical system. So you can take a single point in classical configuration space, and ask how the corresponding system develops over time. You can take a single point in classical configuration space, and ask, "Where does this one point go?"

The development over time of quantum systems depends on things like the second derivative of the amplitude distribution. Our laws of physics describe how amplitude distributions develop into new amplitude distributions. They do not describe, even in principle, how one configuration develops into another configuration.

Instead of viewing the wavefunction as some kind of structure encompassing many points in configuration space, you can view the wavefunction as a whole as a single point in configuration space. Then the evolution in configuration space does indeed depend only on the point itself, not its neighbourhood.