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The way Feynman expresses the flow of amplitude to a certain point given a prior configuration is as a weighted sum over space of sums over path weights. The sum over space is simply weighted by the amplitude distribution of the given configuration and the weight of each path is but itself a sum over time of a quantity called Lagrangian (more precisely the complex exponential of this quantity but whatever) along said path.
Since this quantity is the difference between kinetic and potential energy, it normally should only depends on the position and time derivatives along the path. In that sense the path integral formalism for a finite number of particles is independent of the derivative of the amplitude distribution itself and thus of Schrödinger equation.
If one now goes to a situation with an infinite number of degrees of freedom, that is a field, and tries to implement there also a path integral formalism, then the equation changes slightly. Amplitude doesn't flow from one point to the other but rather between field configurations. In that case the second sum is not over all possible paths in between two points but over all possible field configurations in between two field configurations. Doing so, the quantity used to weight configurations now depends on the amplitude and space derivatives of the field everywhere.
And if one fancies a Schrödinger equation for a quantum field, then in the interacting and non-relativistic case this equation turns out to be nonlocal and nonlinear.