This paper analyses the dynamic aspects of knowledge sharing in R&D rivalry. In a model where research projects consist of N sequential stages, our goal is to explore how the innovators' incentives to share intermediate research outcomes change with progress and with their relative positions in an R&D race. We consider an uncertain research process, where progress implies a decrease in the level of uncertainty that a firm faces. We assume that firms are informed about the progress of their rivals and make joint sharing decisions either before or after each success. Changes in the firms' absolute and relative positions affect their incentives to stay in the race and the expected duration of monopoly profits if they finish the race first. We show that firms always prefer to have sharing between their independent research units if they are allowed to collude in the product market. However, competing firms may have either decreasing or increasing incentives to share intermediate research outcomes throughout the race. If the lagging firm never drops out, the incentives to share always decrease over time as the research project nears completion. The incentives to share are higher earlier on because sharing has a smaller impact on each firm's chance of being a monopolist at the end of the race. If the lagging firm is expected to drop out, the incentives to share may increase over time. We also use our framework to analyze the impact of patent policy on the sharing incentives of firms and show that as patent policy gets stronger, sharing incentives may decrease or increase depending on whether or not the lagging firm has increased incentives to drop out.