Plasma sheet convection is a key element of storm‐time plasma dynamics in the magnetosphere. While decades of observations have advanced our understanding of convection in general, specifically storm‐time convection remains poorly understood. Using data from ISAS/NASA's Geotail and NASA's MMS, this study characterizes plasma sheet magnetic flux transport across the magnetotail during numerous storms (both recovery and main phases) and contrasts these observations with those from quiet times. Our findings confirm the well‐documented enhancement of the convection electric field during geomagnetic storms. Beyond that, our results reveal a significant dawn‐dusk asymmetry. At dawn, the elevated convection is realized via relatively faster flows while at dusk, through a stronger northward magnetic field. These findings suggest a complex feedback loop between plasma sheet convection and ring current buildup, whereby the latter asymmetrically inflates the magnetotail on the dusk side, shifting the reconnection site and subsequently enhanced earthward flows toward dawn.