  We study variable-speed random walks on $\mathbb Z$ driven by a family of
nearest-neighbor time-dependent random conductances $\{a_t(x,x+1)\colon
x\in\mathbb Z, t\ge0\}$ whose law is assumed invariant and ergodic under
space-time shifts. We prove a quenched invariance principle for the random walk
under the minimal moment conditions on the environment; namely, assuming only
that the conductances possess the first positive and negative moments. A novel
ingredient is the representation of the parabolic coordinates and the corrector
via a dual random walk which is considerably easier to analyze.
