We studied the friction between two immiscible polymer melts of polybutadiene (PBD) and polydimethylsiloxane (PDMS). Polymer films (100-300 nm thick) were coated onto smooth mica substrates, and then were brought into contact and sheared (slid) to and fro in a surface forces apparatus (SFA). Stop-wait-start experiments were also carried out at different sliding velocities to investigate the characteristic relaxation times of the interdigitation and disinterdigitation processes at both the static and shearing interfaces. By virtue of their limited interdigitation/interpenetration across the contact interface, immiscible polymers never fully coalesce into a continuous homogeneous material. This affects both their dynamic adhesion and friction forces. The immiscible interface exhibits various "characteristic" parameters such as its static and dynamic widths and at least two relaxation times: the static interpenetration time and the velocity adaptation time. The interfacial width saturates at some small but finite value, resulting in Stribeck-like behavior for the friction force as a function of the sliding velocity, characterized by F having a minimum value at some characteristic sliding velocity V. The presence of solvents at the immiscible interface can have a dramatic effect on the friction or lubrication forces. The implications of the results regarding the depth and dynamics of interdigitation and interpenetration of immiscible chains across an interface are discussed in relation to the adhesion, friction, and strength of polymer composites and the coalescence of immiscible droplets.