It's as easy as 1,2,3 (except you have to add them up)
Jelly, 12 11 bytes

+ɼ0x;R}ʋ/€S


Try it online!

A full program that takes a list of lists of integers as its argument and returns a list of integers. Can be adapted to work as a link by resetting the register to zero after each call (as implemented in the footer on TIO).

Saved a byte now list can be non-empty.