  We prove a law of large numbers for certain random walks on certain
attractive dynamic random environments when initialised from all sites equal to
the same state. This result applies to random walks on $\mathbb{Z}^d$ with
$d\geq1$. We further provide sufficient mixing conditions under which the
assumption on the initial state can be relaxed, and obtain estimates on the
large deviation behaviour of the random walk.
  As prime example we study the random walk on the contact process, for which
we obtain a law of large numbers in arbitrary dimension. For this model,
further properties about the speed are derived.
