  (abridged) Variability is a key property of stars on the asymptotic giant
branch (AGB). Their pulsation period is related to the luminosity and mass-loss
rate (MLR) of the star. Long-period variables (LPVs) and Mira variables are the
most prominent of all types of variability of evolved stars. However, the
reddest, most obscured AGB stars are too faint in the optical and have eluded
large variability surveys. Our goal is to obtain a sample of LPVs with large
MLRs by analysing WISE W1 and W2 light curves (LCs) for about 2000 sources,
photometrically selected to include known C-stars with the 11.3 mu silicon
carbide dust feature in absorption, and Galactic O-stars with periods longer
than 1000 days. Epoch photometry was retrieved from the AllWISE and NEOWISE
database and fitted with a sinus curve. Photometry from other variability
surveys was also downloaded and fitted. For a subset of 316 of the reddest
stars, spectral energy distributions (SEDs) were constructed, and, together
with mid-infrared (MIR) spectra when available, fitted with a dust radiative
transfer programme in order to derive MLRs. WISE based LCs and fits to the data
are presented for all stars. Inspired by a recent paper, a number of
non-variable OH/IRs are identified. Based on a selection on amplitude, a sample
of about 750 (candidate) LPVs is selected of which 145 have periods over 1000
days, many of them being new. For the subset of the stars with the colours of
C-rich extremely red objects (EROs) the fitting of the SEDs (and available MIR
spectra) separates them into C- and O-rich objects. The number of Galactic EROs
appears to be complete up to about 5~kpc and a total dust return rate in the
solar neighbourhood for this class is determined. Based on the EROs in the
Magellanic Clouds, a bolometric period luminosity is derived.
