  We consider a directed random walk on the backbone of the supercritical
oriented percolation cluster in dimensions $d+1$ with $d \ge 3$ being the
spatial dimension. For this random walk we prove an annealed local central
limit theorem and a quenched local limit theorem. The latter shows that the
quenched transition probabilities of the random walk converge to the annealed
transition probabilities reweighted by a function of the medium centred at the
target site. This function is the density of the unique measure which is
invariant for the point of view of the particle, is absolutely continuous with
respect to the annealed measure and satisfies certain concentration properties.
