You should perhaps skim through https://docs.docker.com/storage/ quickly. That document explains that docker containers only have very limited persistence (this is kind of the whole point of containers). The only persistence of note is volumes. This is normally how settings are saved between recreating containers.
As for dependencies, well it's possible that one container depends on the service of another. Perhaps this is what you are describing?
Either way, for more detailed help, you will have to explain your setup with more specific technical details.