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The asymptotic dynamics of many-qubit quantum systems is investigated under iteratively and randomly applied unitary transformations. For a one-parameter family of unitary transformations, which entangle pairs of qubits, two main theorems are proved. They characterize completely the dependence of the resulting asymptotic dynamics on the topology of the interaction graph that encodes all possible qubit couplings. These theorems exhibit clearly which aspects of an interaction graph are relevant and which ones are irrelevant to the asymptotic dynamics. On the basis of these theorems, the local entropy transport between an open quantum system and its environment are explored for strong non-Markovian couplings and for different sizes of the environment and different interaction topologies. It is shown that although the randomly applied unitary entanglement operations cannot decrease the overall entropy of such a qubit network, a local entropy decrease or ‘cooling’ of subsystems is possible for special classes of interaction topologies.