Polymer-layered silicate nanocomposites may be formed by annealing layered silicate particles with a polymer melt. Polymer molecules flow from a bulk melt into the galleries between silicate sheets, swelling the silicate structure. The use of an amphiphilic intercalant raises possibilities of forming novel structures and enhancing the intercalation kinetics relative to the case of homopolymer intercalants. We perform molecular dynamics simulations of the flow of a symmetric diblock copolymer from a bulk melt into a slit whose surfaces are modified by grafted surfactant chains, and whose walls are maintained at a constant pressure to permit the slit to open as polymer intercalates. Intercalation kinetics are examined for a variety of polymer-surface and interblock interactions and for thermodynamic states in which the bulk polymer occupies either a lamellar or disordered phase. Comparison to previous simulations of homopolymer intercalation demonstrates that diblock copolymers may be used to intercalate a block that would not spontaneously intercalate as a homopolymer.