This is approximately right, time-evolution of a spatially homogeneous system is convolution of its initial state against the propagator `K = \int e^{iS[x]} Dx`

This is approximately right, time-evolution of a spatially homogeneous system is convolution of its initial state against the propagator `K = \int e^{iS[x]} Dx` over all of space - with the caveat that none of this is quite true. These aren't really integrals, `Dx` isn't really a measure, and fully rigorous interacting QFTs in 4d may or may not exist.