If you have a playlist that updates frequently, set to download, then it should automatically update the downloads so you can listen to it even when you're offline.
Now, Spotify might look for changes in the background while you use the Spotify app so you don't have to visit the playlist before it downloads the updated songs. I haven't tested this out because I do not download any playlists that change like this frequently. So it might be that you have to first visit the playlist before it will check for updates (not automatic) not really sure about that
If you have turned off downloads over cellular data in the settings then it will only download when you are connected to WiFi. Just in case, I suggest to monitor it just to make sure it is updating the downloads when the playlist changes, and that it is not downloading over cellular data.
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Note: I'm not a Spotify employee.