Changing storage backend for existing mimir installation

How hard is it to change the storage backed of an existing mimir installation without any downtime?

I’m planning to use the filesystem storage method for now. At a later point though we will probably switch to an object storage like S3 or similar. I wonder how easy it is to sync the new storage and if this can be done without any downtime.

In case it’s relevant, mimir will be deployed in monolithic configuration.

I didn’t find a lot about this when searching here, most questions were about making a backup and then restoring from that. But that would disregard the metrics coming in between the backup being made and then being restored at the new location. I’d appreciate any pointers in the right direction. Even just a few keywords that I may have not searched for yet since English isn’t my native language.