Video-grounded dialogues are very challenging due to (i) the complexity of videos which contain both spatial and temporal variations, and (ii) the complexity of user utterances which query different segments and/or different objects in videos over multiple dialogue turns. However, existing approaches to video-grounded dialogues often focus on superficial temporal-level visual cues, but neglect more fine-grained spatial signals from videos. To address this drawback, we propose Bi-directional Spatio-Temporal Learning (BiST), a vision-language neural framework for high-resolution queries in videos based on textual cues. Specifically, our approach not only exploits both spatial and temporal-level information, but also learns dynamic information diffusion between the two feature spaces through spatial-to-temporal and temporal-to-spatial reasoning. The bidirectional strategy aims to tackle the evolving semantics of user queries in the dialogue setting. The retrieved visual cues are used as contextual information to construct relevant responses to the users. Our empirical results and comprehensive qualitative analysis show that BiST achieves competitive performance and generates reasonable responses on a large-scale AVSD benchmark. We also adapt our BiST models to the Video QA setting, and substantially outperform prior approaches on the TGIF-QA benchmark.