We derive some Moore-like bounds for multipartite digraphs, which extend those of bipartite digraphs, under the assumption that every vertex of a given partite set is adjacent to the same number $\delta$ of vertices in each of the other independent sets. We determine when a Moore multipartite digraph is weakly distance-regular. Within this framework, some necessary conditions for the existence of a Moore $r$-partite digraph with interpartite outdegree $\delta>1$ and diameter $k=2m$ are obtained. In the case $\delta=1$, which corresponds to almost Moore digraphs, a necessary condition in terms of the permutation cycle structure is derived. Additionally, we present some constructions of dense multipartite digraphs of diameter two that are vertex-transitive.