Nuclear star clusters (NSCs) are massive star clusters found in all types of galaxies from dwarfs to massive galaxies. Recent studies show that while low-mass NSCs in dwarf galaxies (M-gal< 10(9)M(circle dot)) form predominantly out of the merger of globular clusters (GCs), high-mass NSCs in massive galaxies have assembled most of their mass through central enriched star formation. So far, these results of a transition in the dominant NSC formation channel have been based on studies of early-type galaxies and massive late-type galaxies. Here, we present the first spectroscopic analysis of a sample of nine nucleated late-type dwarf galaxies with the aim of identifying the dominant NSC formation pathway. We use integral-field spectroscopy data obtained with the Multi Unit Spectroscopic Explorer (MUSE) instrument to analyse the ages, metallicities, star formation histories, and star formation rates of the NSCs and their surroundings. Our sample includes galaxies with stellar masses M-gal=10(7) - 10(9)M(circle dot) and NSC masses M-NSC=6 x 10(4) - 6 x 10(6)M(circle dot). Although all NSC spectra show emission lines, this emission is not always connected to star formation within the NSC, but rather to other regions along the line of sight. The NSC star formation histories reveal that metal-poor and old populations dominate the stellar populations in five NSCs, possibly stemming from the inspiral of GCs. The NSCs of the most massive galaxies in our sample show significant contributions from young and enriched populations that indicate additional mass growth through central star formation. Our results support previous findings of a transition in the dominant NSC formation channel with galaxy mass, showing that the NSCs in low-mass galaxies predominantly grow through the inspiral of GCs, while central star formation can contribute to NSC growth in more massive galaxies.