Dynamical properties of an individual diblock copolymer are investigated. A generalized Rouse equation is derived for polymer chains composed of several blocks differing with respect to their monomeric units. This equation takes into account explicitly the dynamics of the block copolymer melt surrounding the probe chain. In general, the structure of this generalized Rouse equation differs from the corresponding results for a multicomponent homopolymer blend. It is applied to diblock copolymers close to the microphase separation, which is characterized by strong composition fluctuations and the critical slowing down of their dynamics. Emphasis is laid on the role of composition fluctuations in the dynamics of individual diblock chains. A slowing down with respect to the center-of-mass motion as well as the internal relaxations is expected. Results for the Gaussian model and the Brazovskii theory are compared. They show significant differences.