We study the asymptotic behaviour of a random walk whose evolution is dependent on the state of an itself dynamically evolving environment. In particular, we extend our previous results in [Bethuelsen and Völlering, 2016] and prove a strong law of large numbers and large deviation estimates assuming that the dynamic environment is "path-cone"-mixing. Under a mild assumption on the decay rate of this mixing property we further obtain a functional central limit theorem under the annealed law. Our method of proofs rest on the study of the so-called local environment process and general results for $φ$-mixing stochastic processes.