Raw video files associated with the paper described as follows: The cellular cytoskeleton relies on diverse populations of motors, filaments, and binding proteins acting in concert to enable non-equilibrium processes ranging from mitosis to chemotaxis. The cytoskeleton’s versatile reconfigurability, programmed by interactions between its constituents, make it a foundational active matter platform. However, current active matter endeavors are limited largely to single force-generating components acting on a single substrate–far from the composite cytoskeleton in cells. Here, we engineer actin-microtubule composites, driven by kinesin and myosin motors and tuned by crosslinkers, to ballistically restructure and flow with speeds that span three orders of magnitude depending on the composite formulation and time relative to the onset of motor activity. Differential dynamic microscopy analyses reveal that kinesin and myosin compete to delay the onset of acceleration and suppress discrete restructuring events, while passive crosslinking of either actin or microtubules has an opposite effect. Our minimal advection-diffusion model and spatial correlation analyses correlate these dynamics to structure, with motor antagonism suppressing reconfiguration and de-mixing, while crosslinking enhances clustering. Despite the rich formulation space and emergent formulation-dependent structures, the non-equilibrium dynamics across all composites and timescales can be organized into three classes–slow isotropic reorientation, fast directional flow, and multimode restructuring. Moreover, our mathematical model demonstrates that diverse structural motifs can arise simply from the interplay between motor-driven advection and frictional drag. These general features of our platform facilitate applicability to other active matter systems, and shed light on diverse ways that cytoskeletal components can cooperate or compete to enable wide-ranging cellular processes.