The problem of matroid-reachability-based packing of arborescences was solved by Király. Here we solve the corresponding decomposition problem that turns out to be more complicated. The result is obtained from the solution of the more general problem of matroid-reachability-based $(\ell,\ell')$-limited packing of arborescences where we are given a lower bound $\ell$ and an upper bound $\ell'$ on the total number of arborescences in the packing. The problem is considered for branchings and in directed hypergraphs as well.