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We leverage electroosmotic-flow generation in porous media in combination with a hydrophobic air gap to create a controllable valve capable of operating in either finite dosing or continuous flow mode, enabling the implementation of multi-step assays on paper-based devices. The hydrophobic air gap between two paper pads creates a barrier keeping the valve nominally closed. Electroosmotic actuation, implemented using a pair of electrodes under the upstream pad, generates sufficient pressure to overcome the barrier and connect the two pads. We present a model describing the flow and governing parameters, including the electric potentials required to open and close the valve and the threshold potential for switching between the modes of operation. We construct the air gap using a hierarchical superhydrophobic surface and study the stability of the closed valve under strenuous conditions and find good agreement between our model and experimental results, as well as stable working conditions for practical applications. We present a straightforward design for a compact and automated device based on paper pads placed on top of printed circuit boards (PCB), equipped with heating and actuation electrodes and additional power and logic capabilities. Finally, we demonstrate the use of the device for amplification of SARS-CoV-2 sequences directly from raw saliva samples, using a loop-mediated isothermal amplification (LAMP) protocol requiring sample lysis followed by enzymatic deactivation and delivery to multiple amplification sites. Since PCB costs scale favorably with mass-production, we believe that this approach could lead to a low-cost diagnostic device that offers the sensitivity of amplification methods. [ABSTRACT FROM AUTHOR]