Complex fluids have microstructures that evolve during flow, and this tends to modify the behavior of the interface in two-component systems. The resulting moving-boundary problems also present greater challenges to numerical computation. As examples of such unusual interfacial dynamics, I will describe two intriguing phenomena: (i) partial coalescence between a drop and an interface for Newtonian and polymeric liquids; (ii) motion and self-assembly of micro-droplets in a nematic liquid crystal. In the former, the dominant mechanism turns out to be the elongational viscosity suppressing capillary breakup. In the latter, the key is how orientational defects around the droplets evolve and mediate the interaction among neighboring droplets. In both cases, we seek to integrate experimental observation and numerical computation for an understanding of the physics.