1. Open a file stored in Dropbox or Google Drive
2. Close file and note that it is listed in the Recent Files list
3. Modify the file from elsewhere (e.g. on another machine) and confirm that the changes have synced to the provider
4. Reopen the file in Orgro from the Recent Files list

With Google Drive (and Dropbox, according to reports) the contents will often not be updated. With iCloud Drive the contents do seem to update consistently.

It seems you can force-update the data if you:

1. Find the file in the picker or the iOS Files app
2. Long-press it and choose Info from the popup context menu
3. Wait for the Information section to populate
4. Then pick the file

It is unknown if this is a bug or is intentional behavior.