  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 $0.5$ - $9.0 \times10^{40}~\mathrm{erg~s}^{-1}$ and
their total radiated energies are on the order of (0.3 - 3)
$\times$~10$^{47}$~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 $^{56}$Co decay. If the late-time power source is indeed
radioactive decay, these transients produce $^{56}$Ni masses on the order of
$10^{-4}$ to $10^{-3}$~\msun. 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] $\lambda$7291,7324 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.
