Aims. We investigate the spectroscopic characteristics of intermediate-luminosity Red Transients (ILRTs), a class of elusive objects with peak luminosity between that of classical novae and standard supernovae. Our goal is to provide a stepping stone in the path to unveiling the physical origin of these events based on the analysis of the collected datasets. Methods. We present the extensive optical and near-infrared (NIR) spectroscopic monitoring of four ILRTs, namely NGC 300 2008OT-1, AT 2019abn, AT 2019ahd and AT 2019udc. First we focus on the evolution of the most prominent spectral features observed in the low-resolution spectra. We then present a more detailed description of the high-resolution spectrum collected for NGC 300 2008OT-1 with the Very Large Telescope equipped with UVES. Finally, we describe our analysis of late-time spectra of NGC 300 2008OT-1 and AT 2019ahd through comparisons with both synthetic and observed spectra. Results. Balmer and Ca lines dominate the optical spectra, revealing the presence of slowly moving circumstellar medium (CSM) around the objects. The line luminosity of Hα, Hβ, and Ca II NIR triplet presents a double peaked evolution with time, possibly indicative of interaction between fast ejecta and the slow CSM. The high-resolution spectrum of NGC 300 2008OT-1 reveals a complex circumstellar environment, with the transient being surrounded by a slow (∼30 km s-1) progenitor wind. At late epochs, optical spectra of NGC 300 2008OT-1 and AT 2019ahd show broad (∼2500 km s-1) emission features at ∼6170 Å and ∼7000 Å which are unprecedented for ILRTs. We find that these lines originate most likely from the blending of several narrow lines, possibly of iron-peak elements.