If you are using Sierra or High Sierra.. and even going back to El Capitan you might well find issues with permissions.
On a network drive.. permissions are a real problem. Apple has removed fixing permissions in the disk utility.. since they claim there are no more problems.. this is clearly fine for local drives.. ie when plugged into your computer but it simply isn't the case on networks.
There are command line permission fixes.. but like all command line stuff... it is super dangerous to use..
My solution would be to simply get another drive.. plug it into the TC.. create a main folder from your new mac.. Then copy all the files across from the old drive plugged directly into your computer.. that will fix the permissions.