The basal ganglia (BG) are a collection of subcortical nuclei involved in motor control, sensorimotor integration, and procedural learning. They play a key role in the acquisition and adaptation of movements, a process driven by dopamine-dependent plasticity at cortico-striatal projections, which serve as BG input. However, BG output is not necessary for executing many well-learned movements. This raises a fundamental question: How can plasticity at BG input contribute to the acquisition and adaptation of movements which execution does not require BG output? Existing models of BG function often neglect the feedback dynamics within cortico-BG-thalamo-cortical circuitry and do not capture the interaction between the cortex and BG in movement generation and adaptation. In this work, we address the above question in a theoretical model of the BG-thalamo-cortical multiregional network, incorporating anatomical, physiological, and behavioral evidence. We examine how its dynamics influence the execution and reward-based adaptation of reaching movements. We demonstrate how the BG-thalamo-cortical network can shape cortical motor output through the combination of three mechanisms: i) the diverse dynamics emerging from its closed-loop architecture, ii) attractor dynamics driven by recurrent cortical connections, and iii) reinforcement learning via dopamine-dependent cortico-striatal plasticity. Our study highlights the role of the cortico-BG-thalamo-cortical feedback in efficient visuomotor adaptation. It also suggests a mechanism for early-stage acquisition of reaching movements through motor babbling. More generally, our model explains how the BG-cortical network refines motor output through its intricate closed-loop dynamics and dopamine-dependent plasticity at cortico-striatal synapses.