On cp3040 /etc/cron.d/varnish-backend-restart failed to properly restart varnish-be. From the logs:
Nov 03 08:20:38 cp3040 varnishd: WARNING: (-spersistent) file size does not fit in reported free space, continuing anyways!
After that, the following message got reported a few times:
Error: (-spersistent): fallocate() for file /srv/sda3/varnish.main1 failed: No space left on device
Restarting the service by hand with service varnish restart solved the issue.
It's worth noting that /usr/local/sbin/varnish-backend-restart stops varnish-be, wipes /srv/sd*/varnish* and then starts varnish-be again.