Lateral transshipments are a method of responding to shortages of stock in a network of inventory-holding locations. Conventional reactive approaches only seek to meet immediate shortages. The study proposes hybrid transshipments which exploit economies of scale by moving additional stock between locations to prevent future shortages in addition to meeting immediate ones. The setting considered is motivated by retailers who operate networks of outlets supplying car parts via a system of periodic replenishment. It is novel in allowing non-stationary stochastic demand and general patterns of dependence between multiple item types. The generality of our work makes it widely applicable. We develop an easy-to-compute quasi-myopic heuristic for determining how hybrid transshipments should be made. We obtain simple characterizations of the heuristic and demonstrate its strong cost performance in both small and large networks in an extensive numerical study.