Magnetic resonance imaging (MRI) can elucidate the interior structure of an optically opaque object in unparalleled detail but is ultimately limited by the need to enclose the object within a detection coil; acquiring the image with increasingly smaller pixels reduces the sensitivity, because each pixel occupies a proportionately smaller fraction of the detector's volume. We developed a technique that overcomes this limitation by means of remotely detected MRI. Images of fluids flowing in channel assemblies are encoded into the phase and intensity of the constituent molecules' nuclear magnetic resonance signals and then decoded by a volume-matched detector after the fluids flow out of the sample. In combination with compressive sampling, we thus obtain microscopic images of flow and velocity distributions ~10(6) times faster than is possible with conventional MRI on this hardware. Our results illustrate the facile integration of MRI with microfluidic assays and suggest generalizations to other systems involving microscopic flow.