git annex export corresponding to import. This might be useful for eg, datalad. There are some requests to make eg a S3 bucket mirror the filenames in the git annex repository with incremental updates, which seem out of scope (and there are many tools to do stuff like that search "deploy files to S3 bucket"), but something simpler like git annex export could be worth doing.

git annex export --to remote files would copy the files to the remote, using the names in the working tree. For remotes like S3, it could add the url of the exported file, so that another clone of the repo could use the exported data.

Would this be able to reuse the existing storeKey interface, or would there need to be a new interface in supported remotes?