For text-to-video retrieval (T2VR), which aims to retrieve unlabeled videos by ad-hoc textual queries, CLIP-based methods are dominating. Compared to CLIP4Clip which is efficient and compact, the state-of-the-art models tend to compute video-text similarity by fine-grained cross-modal feature interaction and matching, putting their scalability for large-scale T2VR into doubt. For efficient T2VR, we propose TeachCLIP with multi-grained teaching to let a CLIP4Clip based student network learn from more advanced yet computationally heavy models such as X-CLIP, TS2-Net and X-Pool . To improve the student's learning capability, we add an Attentional frame-Feature Aggregation (AFA) block, which by design adds no extra storage/computation overhead at the retrieval stage. While attentive weights produced by AFA are commonly used for combining frame-level features, we propose a novel use of the weights to let them imitate frame-text relevance estimated by the teacher network. As such, AFA provides a fine-grained learning (teaching) channel for the student (teacher). Extensive experiments on multiple public datasets justify the viability of the proposed method.