We introduce and analyze a discrete-time network process in which each node holds a (weak) preference ordering over a finite set of alternatives and updates by local Borda aggregation. At each step, a node forms a weighted average (row-stochastic random-walk normalization) of its neighbors' Borda score vectors and projects the aggregated score back to a weak order. Updates are bounded: in each round, a node advances by at most one step along a shortest path in the fixed graph of preference orderings, following the direction prescribed by its neighbors' Borda-aggregated preferences. Our emphasis is dynamical: we develop sufficient conditions, stated directly in terms of graph topology, weights, and the bounded step rule, for (i) self-sustained oscillations in the absence of persistent sources, and (ii) forced oscillations under contrarian persistent camps. We also record robustness (structural stability) away from score-tie hyperplanes and contrast synchronous (Variant S) and asynchronous (Variant A) updating.