According to the posting Mr. V pointed out, you have 3 moodle sites - each has it's own unique moodledata diretory. Your screen shot shows a listing of the 0 byte log files of just one of those. Again, 0 byte log files are an indication that backups completed successfully - so one of your three sites is ok.
You mention 'given destination' .... that's a path .... which are/is?
If you are still running 3 sites, each moodle site's config for autobackups would have it's own unique destination path.
In all three sites, there is a report for backups:
/report/backups/index.php
What does each site report show?
In the setup of autobackups one has options for when to backup ... for example, if there has been no change to a course, no backup is made.
I take it that all 3 sites cron jobs are running as they should with no errors?
In this thread you say you have a 4.0.1 (so that's now 4 sites you are running?). That's already behind 2 point releases + ... and there was just recently an 'out of band' (not normal release date) for 4.0 due to some security issue. So advise you update as well.
Also, you are running autobackups as a separate cron job, I see. Reason for that? In the config of auto, is it set to manual?
'SoS', Ken