This is an answer to my own question.
I found the following pure Docker approach. I was looking for a way to copy files to a volume directly, but that does not seem to be supported for some reason?
You can copy files to a container however. So the following works:
export DOCKER_HOST=tcp://remote_host:2375
docker volume create data-volume
docker create -v data-volume:/data --name helper busybox true
docker cp /home/Bradson/data helper:/data
docker rm helper
docker run -v data-volume:/data a
This is inspired by https://stackoverflow.com/a/37469637/10042924.