In a recent paper, DeBacker and Reeder have constructed a piece of the local Langlands correspondence for pure inner forms of unramified $p$-adic groups and have shown that the corresponding L-packets are stable. In this talk we are going to discuss the endoscopic transfer of these L-packets: the theory of endoscopy -- an instance of the broad principle of functoriality -- predicts precise relationships between the L-packets on a group $G$ and its endoscopic groups $H$. This relationship is encoded in the so called endoscopic character identities. We will motivate and state these identities, paying attention to the precise normalization of all objects involved. If time permits, we will then discuss their proof and the extension of the correspondence to non-pure inner forms via the theory of isocrystals with additional structure.