In this article, we prove a lower bound for the fluctuations of symmetric random walks on dynamic random environments in dimension $1 + 1$ in the perturbative regime where the walker is weakly influenced by the environment. We suppose that the random environment is invariant with respect to translations and reflections, satisfies the FKG inequality and a mild mixing condition. The techniques employed are inspired by percolation theory, including a Russo-Seymour-Welsh (RSW) inequality. To exemplify the generality of our results, we provide two families of fields that satisfy our hypotheses: a class of Gaussian fields and Confetti percolation models.