Urban consolidation centers provide the logistical infrastructure for cooperation among less-than-truckload carriers with contiguous destinations. The rising number of initiatives to establish and operate urban consolidation centers and their low success rates signal the need for better mechanisms to manage cooperation in this context. We introduce and study cooperative situations comprising a set of carriers with time sensitive deliveries who can consolidate their cargo to obtain savings. We introduce the class of Dispatch Consolidation (DC) games and search for ways to fairly allocate the obtained savings among the participating carriers. When delivery capacities are not restrictive, i.e. when waiting costs trigger truck dispatches, we show that stable allocations in the core always exist and can, in their entirety, be found by solving a compact linear program. With restrictive capacities, however, the core of a DC game may become empty. We introduce the notion of component-wise core for DC games to preserve stability first and foremost among the carriers whose deliveries are dispatched together in the chosen optimal solutions. The novelty of our approach is to link the stability requirements of an allocation rule with the structure of selected solutions for the underlying optimization problems. We characterize the component-wise cores of DC games, prove their non-emptiness, and suggest proportionally calculated allocations therein. Finally, we discuss a refinement of component-wise core allocations that minimizes envy among the carriers who are dispatched separately.