  We study the typical entanglement properties of a system comprising two
independent qubit environments interacting via a shuttling ancilla. The initial
preparation of the environments is modeled using random-matrix techniques. The
entanglement measure used in our study is then averaged over many histories of
randomly prepared environmental states. Under a Heisenberg interaction model,
the average entanglement between the ancilla and one of the environments
remains constant, regardless of the preparation of the latter and the details
of the interaction. We also show that, upon suitable kinematic and dynamical
changes in the ancilla-environment subsystems, the entanglement-sharing
structure undergoes abrupt modifications associated with a change in the
multipartite entanglement class of the overall system's state. These results
are invariant with respect to the randomized initial state of the environments.
