Multiple transitivity except for a system of imprimitivity

Let $\mathrm{\Omega }$ be a set equipped with an equivalence relation $\sim$; we refer to the equivalence classes as blocks of $\mathrm{\Omega }$. A permutation group $G\le \mathrm{S}\mathrm{y}\mathrm{m}(\mathrm{\Omega })$ is $k$-by-block-transitive if $\sim$ is $G$-invariant, with at least $k$ blocks, and $G$ is transitive on the set of $k$-tuples of points such that no two entries lie in the same block. The action is block-faithful if the action on the set of blocks is faithful. In this article we classify the finite block-faithful $2$-by-block-transitive actions. We also show that for $k\ge 3$, there are no finite block-faithful $k$-by-block-transitive actions with nontrivial blocks.