[whpalmer4]

Unlike what happens on the Mac, you aren't actually opening the original file from the old app's sandbox in the new app, you're opening a copy of it that is made by the OS in the new app's sandbox. Further evidence of this is the fact that if you go change the version in Dropbox, when you go to OmniOutliner you won't find the change there, unless you first go to Dropbox, find the file, and do "Open in OmniOutliner" again. As long as you realize this, and don't mind throwing out old, unneeded versions, this behavior isn't necessarily a bad thing. When you send the document to Dropbox, they give you the warning about overwriting an existing file, whereas Omni just makes a different version. Pretty safe bet that some people will be confused with either option, but the Omni approach at least saves all of the pieces.

There are apps which connect directly to Dropbox's servers and fetch/save the files where you would get the changes propagated to you. They can only do so when you have internet connectivity. Files you've saved to Dropbox and marked as a favorite in the Dropbox app, on the other hand, are cached locally on your device, and you can still access them without internet connectivity, though they might be stale if the document has been changed elsewhere while you've been without connectivity.