Latent dynamical models of the primary motor cortex (M1) have revealed fundamental neural computations underlying motor control; however, such models often overlook the impact of sensory feedback, which can continually update cortical dynamics and correct for external perturbations. This suggests a critical need to model the interaction between sensory feedback and intrinsic dynamics. Such models would also benefit the design of brain-computer interfaces (BCIs) that decode neural activity in real time, where both user learning and proficient control require feedback. Here we investigate the flexible feedback modulation of cortical dynamics and demonstrate its impact on BCI task performance and short-term learning. By training recurrent network models with real-time sensory feedback on a simple 2D reaching task, analogous to BCI cursor control, we show how previously reported M1 activity patterns can be reinterpreted as arising from feedback-driven dynamics. Next, by incorporating adaptive controllers upstream of M1, we make a testable prediction that short-term learning for a new BCI decoder is facilitated by plasticity of inputs to M1, including remapping of sensory feedback, beyond the plasticity of recurrent connections within M1. This input-driven dynamical structure also determines the speed of adaptation and learning outcomes, and explains a continuous form of learning variability. Thus, our work highlights the need to model input-dependent latent dynamics for motor control and clarifies how constraints on learning arise from both the statistical characteristics and the underlying dynamical structure of neural activity.