Variations in the nucleosynthetic isotope compositions of meteorites have been suggested to result from variable destruction of some presolar grains through thermal processing. This mechanism has furthermore been linked to volatile depletion in chondritic meteorites. As a moderately volatile element with variable nucleosynthetic isotope compositions, Zn potentially records both processes. Here, we present the nucleosynthetic Zn isotope compositions of leachates that target distinct mineral phases from four carbonaceous chondrites (CCs). The isotopic similarities between some leaching steps imply that different phases were partially homogenized, obscuring the original hosts of the neutron-rich (NR) Zn component that characterizes CCs. However, some leachates of the most primitive sample, Murchison CM2, display distinct Zn isotope compositions. The results reveal that the NR Zn was possibly hosted in silicate or oxide grains, which are similar or identical to the carriers of the bulk of the Zn. Multiple scenarios that simulate preferential Zn loss from such phases were modelled and compared to estimated compositions of thermally processed components in chondrites. The modelling suggests that variable extents of Zn depletion cannot account for the observed variations in nucleosynthetic Zn isotope compositions, implying that these characteristics are decoupled. Instead, distinct isotope compositions are required for the different Zn carriers depending on the chondrite group. The results thus suggest that thermal processing alone is not responsible for the observed nucleosynthetic Zn isotope variations. Rather, they most likely reflect temporal or spatial changes in the composition of the molecular cloud as the protoplanetary disk evolved, in line with other studies.