I think these are the steps for updating data for your Tableau Reader users:
Step 1. Update the file in Tableau Desktop (if .twb, then save as .twbx)
Step 2. Upload .twbx file to Dropbox (either by overwriting the previous or as a new file)
Step 3. Next update: Repeat step 1 and 2.
I think Reader is designed to be costly to distribute to avoid decrease in sales of Server/Desktop licenses. The day someone shares an effective solution to automatically distribute .twbx files is probably the day Tableau stops offering Reader or begins to charge a license fee for it.