In the Inventory Routing Problem customer demand is satisfied from inventory which is replenished with capacitated vehicles. The objective is to minimize total routing and inventory holding cost over a time horizon. If the customers are located relatively close to each other, one has the opportunity to satisfy a part of the demand of a customer by inventory stored at another nearby customer. In the optimization of the customer replenishments, this option can be included to lower total inventory routing costs. This is for example the case for ATMs in urban areas where an ATM-user, that wants to withdraw money, could be redirected to another ATM. To the best of our knowledge, the possibility of redirecting end-users is new to the operations research literature and has not been implemented, but is being considered, in the industry. We formulate the Inventory Routing Problem with Demand Moves in which demand of a customer can (partially) be satisfied by the inventory of a nearby customer at a service cost depending on the quantity and the distance. We propose a branch-price-and-cut solution approach which is evaluated on problem instances derived from the literature. Cost improvements over the classical IRP of up to 10% are observed with average savings around 3%.