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The cytoskeleton of biological cells relies on a diverse population 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 live cells. Here, we engineer actin-microtubule composites, driven by kinesin and myosin motors and tuned by crosslinkers, that restructure into diverse morphologies from interpenetrating filamentous networks to de-mixed amorphous clusters. Our Fourier analyses reveal that kinesin and myosin compete to delay kinesin-driven restructuring and suppress de-mixing and flow, while crosslinking accelerates reorganization and promotes actin-microtubule correlations. The phase space of non-equilibrium dynamics falls into three broad classes– slow reconfiguration, fast advective flow, and multi-mode ballistic dynamics – with structure-dynamics relations described by the relative contributions of elastic and dissipative responses to motor-generated forces.