We study the spin-transfer drag mediated by the Brownian motion of skyrmions. The essential idea is illustrated in a two-terminal geometry, in which a thin film of a magnetic insulator is placed in between two metallic reservoirs. An electric current in one of the terminals pumps topological charge into themagnet via a spin-transfer torque. The charge diffuses over the bulk of the system as stable skyrmion textures. By Onsager's reciprocity, the topological charge leaving the magnet produces an electromotive force in the second terminal. The voltage signal decays algebraically with the separation between contacts, in contrast to the exponential suppression of the spin drag driven by nonprotected excitations like magnons. We show how this topological effect can be used as a tool to characterize the phase diagram of chiral magnets and thin films with interfacial Dzyaloshinskii-Moriya interactions.