Summary: This paper is a follow-up contribution to our paper V. M. Kadets, R. V. Shvidkoy and D. Werner, [Studia Math. 147, 269-298 (2001; Zbl 0986.46010)] where we defined and investigated narrow operators on Banach spaces with the Daugavet property. We characterise narrow and strong Daugavet operators on \(C(K,E)\)-spaces; these are in a way the largest sensible classes of operators for which the norm equation \(\|\text{Id}+T \|=1+ \|T\|\) is valid. For certain separable range spaces \(E\), including all finite-dimensional spaces and all locally uniformly convex spaces, we show that an unconditionally pointwise convergent sum of narrow operators on \(C(K,E)\) is narrow. This implies, for instance, the known result that these spaces do not have unconditional FDDS. In a different vein, we construct two narrow operators on \(C([0,1],\ell_1)\) whose sum is not narrow.