The cytoskeleton dynamically tunes its mechanical properties by altering the interactions between semiflexible actin filaments, rigid microtubules, and crosslinking proteins. Here, we use optical tweezers microrheology and confocal microscopy to characterize how varying crosslinking motifs impact the microscopic and mesoscale mechanics and mobility of actin-microtubule composites. We show that, upon subtle changes in the crosslinking pattern, composites separate into two distinct classes of force responseprimarily elastic versus more viscous behavior. For example, a composite in which actin and microtubules are crosslinked to each other is markedly more elastic than one in which both filaments are crosslinked but cannot link together. Notably, this distinction only emerges at mesoscopic scales in response to nonlinear forcing, whereas varying crosslinking motifs have little impact on the microscale mechanics and steadystate mobility of composites. Our unexpected scale-dependent results not only inform the physics underlying key cytoskeleton processes and structures, but, more generally, provide valuable perspective to materials engineering endeavors focused on polymer composites.