Here, we report efficient composition-tunable Cu-doped ZnInS/ZnS (core and core/shell) colloidal nanocrystals (CNCs) synthesized by using a colloidal non-injection method. The initial precursors for the synthesis were used in oleate form rather than in powder form, resulting in a nearly defect-free photoluminescence (PL) emission. The change in Zn/In ratio tunes the percentage incorporation of Cu in CNCs. These highly monodisperse Cu-doped ZnInS CNCs having variable Zn/In ratios possess peak emission wavelength tunable from 550 to 650 nm in the visible spectrum. The quantum yield (QY) of these synthesized Cd-free CNCs increases from 6.0 to 65.0% after coating with a ZnS shell. The CNCs possessing emission from a mixed contribution of deep trap and dopant states to only dominant dopant-related Stokes-shifted emission are realized by a careful control of stoichiometric ratio of different reactant precursors during synthesis. The origin of this shift in emission was understood by using steady state and time-resolved fluorescence (TRF) spectroscopy studies. As a proof-of-concept demonstration, these blue excitable Cu-doped ZnInS/ZnS CNCs have been integrated with commercial blue LEDs to generate white-light emission (WLE). The suitable combination of these highly efficient doped CNCs results led to a Commission Internationale de l’Enclairage (CIE) color coordinates of (0.33, 0.31) at a color coordinate temperature (CCT) of 3694 K, with a luminous efficacy of optical radiation (LER) of 170 lm/Wopt and a color rendering index (CRI) of 88.Electronic supplementary materialThe online version of this article (10.1186/s11671-018-2599-x) contains supplementary material, which is available to authorized users.