Thanks for your prompt reply. I have not updated or added any new plugins or WordPress versions recently, by which I mean over a month. This all began the last day or so, and I have had the backup plugin long enough to have seen such behavior on previous updates, if that triggered it. Also, between the time I first wrote and now the backup directory went from 15MB to around 14MB (from 5MB when I first started noticing the problem) in the span of a few hours. Meanwhile I was still getting messages telling me hundreds of files were being replaced.
For the time being I’ve deactivated the plugin, which has made it stop.