We present the spectroscopic and photometric study of five intermediate-luminosity red transients (ILRTs), namely AT 2010dn, AT 2012jc, AT 2013la, AT 2013lb, and AT 2018aes. They share common observational properties and belong to a family of objects similar to the prototypical ILRT SN 2008S. These events have a rise time that is less than 15 days and absolute peak magnitudes of between −11.5 and −14.5 mag. Their pseudo-bolometric light curves peak in the range 0.5–9.0 × 1040 erg s−1 and their total radiated energies are on the order of (0.3–3) × 1047 erg. After maximum brightness, the light curves show a monotonic decline or a plateau, resembling those of faint supernovae IIL or IIP, respectively. At late phases, the light curves flatten, roughly following the slope of the 56Co decay. If the late-time power source is indeed radioactive decay, these transients produce 56Ni masses on the order of 10−4 to 10−3 M⊙. The spectral energy distribution of our ILRT sample, extending from the optical to the mid-infrared (MIR) domain, reveals a clear IR excess soon after explosion and non-negligible MIR emission at very late phases. The spectra show prominent H lines in emission with a typical velocity of a few hundred km s−1, along with Ca II features. In particular, the [Ca II] λ7291,7324 doublet is visible at all times, which is a characteristic feature for this family of transients. The identified progenitor of SN 2008S, which is luminous in archival Spitzer MIR images, suggests an intermediate-mass precursor star embedded in a dusty cocoon. We propose the explosion of a super-asymptotic giant branch star forming an electron-capture supernova as a plausible explanation for these events.