We present Lumosaic, a compact active hyperspectral video system designed for real-time capture of dynamic scenes. Our approach combines a narrowband LED array with a coded-exposure-pixel (CEP) camera capable of high-speed, per-pixel exposure control, enabling joint encoding of scene information across space, time, and wavelength within each video frame. Unlike passive snapshot systems that divide light across multiple spectral channels simultaneously and assume no motion during a frame's exposure, Lumosaic actively synchronizes illumination and pixel-wise exposure, improving photon utilization and preserving spectral fidelity under motion. A learning-based reconstruction pipeline then recovers 31-channel hyperspectral (400-700 nm) video at 30 fps and VGA resolution, producing temporally coherent and spectrally accurate reconstructions. Experiments on synthetic and real data demonstrate that Lumosaic significantly improves reconstruction fidelity and temporal stability over existing snapshot hyperspectral imaging systems, enabling robust hyperspectral video across diverse materials and motion conditions.