Companion reflects what the Basecamp public API allows you to do. Two common Finder gestures don't have an equivalent in that API, so Companion doesn't fake them or force them (that would destroy the file's history, comments, and mentions). Instead, Finder reverts the change.
Edit and save over the same file
If you open a Companion file, edit it, and hit Save (for example a .txt in TextEdit or a screenshot in Preview), the app will show a save error by default. Basecamp doesn't have a public way to replace the content of an existing file while preserving its identity.
What to do: save your edited version as a new file and drag it to the folder to upload it. It appears as a new version alongside the original.
Prefer it to just work? In Settings → Docs & Files, turn on Sync edits as new versions (marked Experimental). With it on, saving over a Companion file uploads your edit as a new version on Basecamp instead of failing — the previous version is archived and linked with a comment, and the file keeps its place in Finder. It's off by default because it's still experimental.
Move a file between folders within Companion
If you drag a file from one folder to another within Companion, Finder will move it back. Basecamp doesn't have a public endpoint to change the folder of an already-uploaded file. There's no setting that changes this one.
What to do: download (or copy) the file, upload it to the destination folder, and archive the original if you no longer need it.
Note: declining these by default is intentional, to protect your files' history, comments, and mentions in Basecamp — that's why saving over a file is opt-in rather than the default, and why moving between folders has no equivalent at all. Renaming, uploading, downloading, deleting/archiving, and creating folders all work normally.