Critical minerals are essential to the future of clean energy, especially energy storage, electric vehicles, and advanced electronics. In this paper, we argue that process systems engineering (PSE) paradigms provide essential frameworks for enhancing the sustainability and efficiency of critical mineral processing pathways. As a concrete example, we review challenges and opportunities across material-to-infrastructure scales for process intensification (PI) with membranes. Within critical mineral processing, there is a need to reduce environmental impact, especially concerning chemical reagent usage. Feed concentrations and product demand variability require flexible, intensified processes. Further, unique feedstocks require unique processes (i.e., no one-size-fits-all recycling or refining system exists). Membrane materials span a vast design space that allows significant optimization. Therefore, there is a need to rapidly identify the best opportunities for membrane implementation, thus informing materials optimization with process and infrastructure scale performance targets. Finally, scale-up must be accelerated and de-risked across the materials-to-process levels to fully realize the opportunity presented by membranes, thereby fostering the development of a circular economy for critical minerals. Tackling these challenges requires integrating efforts across diverse disciplines. We advocate for a holistic molecular-to-systems perspective for fully realizing PI with membranes to address sustainability challenges in critical mineral processing. The opportunities for PI with membranes are excellent applications for emerging research in machine learning, data science, automation, and optimization.