I have a directory in my home folder for all my different docker containers. I back that up like my other critical files on my NAS, Backup NAS, cloud storage and external drive. Its all scripted to happen periodically.
The docker compose files I also upload to my local Git so I can version them. Make the git ignore skip everything that isn't a .yml
If you would want to spin them up again, yeah you could just copy them in a new place, make sure the docker-compose.yml is at the top and docker-compose pull the image, then docker-compose up. If your yml specifies that your configs are ./ , it'll know to look for them in the same folder as your yml and begin its file paths there. Anything outside your ./ in theory should be read only or be some kind of data off a NAS that is independent from the service. Its just not best practice in my eyes for your docker application to write outside of the folders beneath its docker-compose.yml location.
Just organize your volumes correctly and you can move the docker container anywhere you want with a copy paste and a docker-compose up. Imo docker-compose is WAY easier and faster than any GUI instance of Docker.