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Recent advances in material synthesis made it possible to realize two-dimensional monolayers of candidate materials for a quantum spin liquid (QSL) such as $α$-RuCl$_3$,1T-TaSe$_2$ and 1T-TaS$_2$. In this work, we propose an experimental setup that exploits nonlocal electrical probes to gain information on the transport properties of a gapless QSL. The proposed setup is a spinon-mediated drag experiment: a current is injected in one of the two layers and a voltage is measured on the second metallic film. The overall momentum transfer mechanism is a two-step process mediated by Kondo interaction between the local moments in the quantum spin liquid and the spins of the electrons. In the limit of negligible momentum relaxed within the QSL layer, we calculate the drag relaxation rate for Kitaev, $\mathbb{Z}_2$, and U(1) QSLs using Aslamazov-Larkin diagrams. We find, however, that the case of dominant momentum relaxation within the QSL layer is far more relevant and thus develop a model based on the Boltzmann kinetic equation to describe the proposed setup. Within this framework we calculate the low temperature scaling behavior of the drag resistivity, both for U(1) and $\mathbb{Z}_2$ QSLs with Fermi surfaces. In some regimes we find a crossover in the temperature scaling that is different between the $\mathbb{Z}_2$ and U(1) QSL both because of the non-Fermi liquid nature of the U(1) QSL, and because of the qualitatively different momentum relaxation mechanism within the QSL layer. Our findings suggest that parameters of the system can be tuned to make the spinon-mediated drag a significant fraction of the total transresistance.