Orbits and their closures in the spaces $\mathbb C^{k_1}\otimes\dots\otimes\mathbb C^{k_r}$

Natural actions of direct products of general linear groups on tensor products of the corresponding complex linear spaces are considered. Among these actions, all actions with finitely many orbits are distinguished. The main results of the paper are the classification of orbits and the construction of the orbit abutment graphs for all such actions.