This app was mentioned in 3 comments, with an average of 4.00 upvotes
I'm looking for the same thing, and haven't had any luck!
I use a variation of your Dropbox method. I manually search for the books, then once downloaded Calibre snags the file and moves it to a library stored in a Dropbox folder. Calibre acts as a server, so can be accessed by many apps (Nook, Kindle, MoonReader, Aldiko, etc.) from outside your network. It takes some elbow grease to get the outside connectivity working properly, but once running does quite a good job.
I am currently having some issues getting past my router, so I found CalibreBox, an Android app that can find and parse your Calibre library in Dropbox. CalibreBox downloads the book files to the device where the reader apps can then add them to their individual libraries. I'm pretty much exclusively Android, so this solution will work for me until I can find a better method.
I wish there was a Sonarr/Plex-type solution for books!
CalibreBox, it allows you to open calibre eBook libraries from Dropbox and search/filter/download ebooks from it.
Store link if you are interested: https://play.google.com/store/apps/details?id=com.hoffmann.eric.calibrebox