Some files in Time Machine are being locked
I just tried "Reset Home Folder Permissions and ACLs" and it looks like it might be helping. I ran this for about 30 minutes, canceled it because it took too long, backed up Time Machine again, and noticed that Time Machine backed up some files that I did not touch. Turns out that these files were the same ones being locked in backups, and in the latest backup, they were no longer locked.

So perhaps I was having permission problems with the files that were getting locked? I don't know if this "fix" will resolve this permanently, or just for now, but it certaily seems to be related.