  We study biased random walks on dynamical percolation on $\mathbb{Z}^d$. We
establish a law of large numbers and an invariance principle for the random
walk using regeneration times. Moreover, we verify that the Einstein relation
holds, and we investigate the speed of the walk as a function of the bias.
While for $d=1$ the speed is increasing, we show that in general this fails in
dimension $d \geq 2$. As our main result, we establish two regimes of
parameters, separated by an explicit critical curve, such that the speed is
either eventually strictly increasing or eventually strictly decreasing. This
is in sharp contrast to the biased random walk on a static supercritical
percolation cluster, where the speed is known to be eventually zero.
