The ongoing energy transition revolves around the utilization of strategic minerals at unprecedented scales. Elements like lithium, predominantly extracted from brines of geological origin, are increasingly demanded for the electrification of the economy. Current extraction methods require the pre-concentration of the mineral, which is achieved in multi-stage ponds where ambient evaporation extracts water from the brine, precipitating the most abundant salts (e.g. sodium chloride) while the target species (lithium) gets concentrated. However, this process has long operating times, requires extensive land use, and depletes surrounding aquifers due to uncontrolled evaporation. Here, we demonstrate how porous cellulose crystallizers can, through evaporation-driven capillary flow, increase the rate of lithium extraction per unit of surface area by an order of magnitude, thereby reducing land use and operation times by over 90% [Chen et al., Nature Water (2023)]. A simple transport model recapitulates the observed high lithium selectivity, which stems from the sharp salt gradients at the top of the evaporator that result from the interplay between advection and diffusion. Motivated by these results, we also explore pathways to enhance mineral extraction through changes in evaporator geometry, and discuss alternative configurations that rely on photocatalytic reactions (rather than crystallization) to passively capture other heavy metal contaminants from water through this evaporation-driven, capillarity-mediated mechanism.