The understanding of fluid transport in miniaturized flow devices is an important component in the design of flow cells, micromixers, and microreactors. In this manuscript, we employ NMR in the form of a voxel-selective multiple modulation multiple echo sequence (MMMEV) to monitor average velocities in individual microchannels inside a six-channel network. The technique produces average velocities which are consistent with the imposed flow rates. In addition, we take advantage of the short acquisition time (32 ms per velocity component) of the technique to quantitatively track the time evolution of the fluid velocity in a pulsatile flow phantom.