The system drive was copied from the old drive by Carbon Copy Cloner. Because the new drive (an SSD) is slightly smaller than the original, CCC could not do a block image copy. Whatever it did do, left the resulting file system in good shape according to Disk Utility and TechTool Pro 5, but with something in there that caused Time Machine to "Process" 370,000 items, and to take three hours to back up a 20Mb incremental backup. I have no idea what Time Machine calls an "item", and Apple being Apple, there's no way to find out.
I had to abandon that attempt, and use Disk Utility to copy the drive. It did do a block copy, since there was enough free space on the source partition for it to get away with doing that. That copy works fine with Time Machine, so the problem's solved.