A novel laboratory experimental design is described that will investigate the processing of dust grains in astrophysical shocks. Dust is a ubiquitous ingredient in the interstellar medium (ISM) of galaxies; however, its evolutionary cycle is still poorly understood. Especially shrouded in mystery is the efficiency of grain destruction by astrophysical shocks generated by expanding supernova remnants. While the evolution of these remnants is fairly well understood, the grain destruction efficiency in these shocks is largely unknown. The experiments described herein will fill this knowledge gap by studying the dust destruction efficiencies for shock velocities in the range ${\sim}10{-}30~\text{km}/\text{s}$ ($\unicode[STIX]{x03BC}\text{m}/\text{ns}$), at which most of the grain destruction and processing in the ISM takes place. The experiments focus on the study of grain–grain collisions by accelerating small (${\sim}1~\unicode[STIX]{x03BC}\text{m}$) dust particles into a large (${\sim}5{-}10~\unicode[STIX]{x03BC}\text{m}$ diameter) population; this simulates the astrophysical system well in that the more numerous, small grains impact and collide with the large population. Facilities that combine the versatility of high-power optical lasers with the diagnostic capabilities of X-ray free-electron lasers, e.g., the Matter in Extreme Conditions instrument at the SLAC National Accelerator Laboratory, provide an ideal laboratory environment to create and diagnose dust destruction by astrophysically relevant shocks at the micron scale.