What are the requirements for supporting non-metallic cable through a finished wall?
You should be fine.  334.30(B)(1) allows you to leave the cable unsupported where it's fished through a finished part of the building.  Which is what you're doing.

334.30(B)(2) removes the requirement to support the cable within 12" of an enclosure, as long as you're within 4 1/2' of the last point of support, you're connecting to a luminaire or other electrical equipment, and the cable and point of connection are within an accessible ceiling in a residential dwelling.