Google Drive downloads - excessive, unwanted.

Are you talking about documents that you made available offline? Open the Drive app, tap Menu>Offline, tap the 3-dot menu button associated with each document that's been made available offline, and then tap "Available offline" to toggle that off.
 
Are you talking about documents that you made available offline? Open the Drive app, tap Menu>Offline, tap the 3-dot menu button associated with each document that's been made available offline, and then tap "Available offline" to toggle that off.

What like they're downloading the same file again and again? I hadn't considered that angle, but in that case they should get to know their little friend File Manager.
 
What like they're downloading the same file again and again? I hadn't considered that angle, but in that case they should get to know their little friend File Manager.

It'd help if the OP clarified what they mean by "drafts." If they're talking about intermittent downloading icons in the status bar related to Google Drive (or one of the Drive apps like Docs, Sheets, etc.), then I figured it's due to offline documents -- I've seen this happen intermittently, probably due to Drive refreshing the data.