Composed Image Retrieval (CoIR) has recently gained popularity as a task that considers both text and image queries together, to search for relevant images in a database. Most CoIR approaches require manually annotated datasets, comprising image-text-image triplets, where the text describes a modification from the query image to the target image. However, manual curation of CoIR triplets is expensive and prevents scalability. In this work, we instead propose a scalable automatic dataset creation methodology that generates triplets given video-caption pairs, while also expanding the scope of the task to include composed video retrieval (CoVR). To this end, we mine paired videos with a similar caption from a large database, and leverage a large language model to generate the corresponding modification text. Applying this methodology to the extensive WebVid2M collection, we automatically construct our WebVid-CoVR dataset, resulting in 1.6 million triplets. Moreover, we introduce a new benchmark for CoVR with a manually annotated evaluation set, along with baseline results. We further validate that our methodology is equally applicable to image-caption pairs, by generating 3.3 million CoIR training triplets using the Conceptual Captions dataset. Our model builds on BLIP-2 pretraining, adapting it to composed video (or image) retrieval, and incorporates an additional caption retrieval loss to exploit extra supervision beyond the triplet, which is possible since captions are readily available for our training data by design. We provide extensive ablations to analyze the design choices on our new CoVR benchmark. Our experiments also demonstrate that training a CoVR model on our datasets effectively transfers to CoIR, leading to improved state-of-the-art performance in the zero-shot setup on the CIRR, FashionIQ, and CIRCO benchmarks. Our code, datasets, and models are publicly available at https://imagine.enpc.fr/ ventural/covr.