This work presents Recurrence with Correlation Network (RWCNet), a novel multi-scale recurrent neural network architecture for medical image registration that integrates core principles from optical flow, including correlation volume computation and inference-time instance optimization. In evaluations on the large-displacement National Lung Screening Test (NLST) dataset, which features large displacements, RWCNet exhibited superior performance (total registration error (TRE) of 2.11mm) to deep learning alternatives, and on par results with variational optimization techniques. In contrast, on the OASIS dataset characterized by smaller displacements, RWCNet's results (average dice similarity of 81.7\%) were superior to variational optimization techniques and showed a small improvement over other multi-scale deep learning models. Ablation experiments showed that multi-scale features consistently improved performance, where as the correlation volume, number of recurrent steps, and inference-time instance optimization only had large positive impacts on performance in the NLST dataset. The performance of RWCNet compared to approaches that use instance optimzation show that deep learning based methods can find local minima that escape instance optimization methods. The results highlight the need for algorithm hyperparameter selection that adjusts with the dataset characteristics. RWCNet's promising results may imporve registration performance and the speed of computation, allowing many potential applications including, treatment planning, intra-procedural guidance, and longitudinal monitoring.