In neuroscience, recurrent neural networks (RNNs) are modeled as continuous-time dynamical systems to more accurately reflect the dynamics inherent in biological circuits. However, convolutional neural networks (CNNs) remain the preferred architecture in vision neuroscience due to their ability to efficiently process visual information, which comes at the cost of the biological realism provided by RNNs. To address this, we introduce a hybrid architecture that integrates the continuous-time recurrent dynamics of RNNs with the spatial processing capabilities of CNNs. Our models preserve the dynamical characteristics typical of RNNs while having comparable performance with their conventional CNN counterparts on benchmarks like ImageNet. Compared to conventional CNNs, our models demonstrate increased robustness to noise due to noise-suppressing mechanisms inherent in recurrent dynamical systems. Analyzing our architecture as a dynamical system is computationally expensive, so we develop a toolkit consisting of iterative and dimensionality-reduction methods specifically tailored for convolutional structures. We also train multi-area RNNs using our architecture as the front-end to perform complex cognitive tasks previously impossible to learn or achievable only with oversimplified stimulus representations. In monkey neural recordings, our models capture time-dependent variations in neural activity in higher-order visual areas. Together, these contributions represent a comprehensive foundation to unify the advances of CNNs and dynamical RNNs in vision neuroscience.