  In this paper, we study random walks evolving on Z in a dynamic random
environment that we assume to have time correlations that decrease polynomially
fast. We show a law of large numbers by generalizing methods already used for
the nearest-neighbor framework to the finite-range one. This requires some new
ideas to get around the absence of a monotonicity property that was crucial in
the proof for the nearest-neighbour case. Our proof works both in discrete and
continuous time.
