  As a result of competing physical mechanisms, the atmospheric composition of
white dwarfs changes throughout their evolution, a process known as spectral
evolution. Because of the ambiguity of their atmospheric compositions and the
difficulties inherent to the modeling of their dense atmospheres, no consensus
exists regarding the spectral evolution of cool white dwarfs ($T_{\rm
eff}<6000$ K). In the previous papers of this series, we presented and
observationally validated a new generation of cool white dwarf atmosphere
models that include all the necessary constitutive physics to accurately model
those objects. Using these new models and a homogeneous sample of 501 cool
white dwarfs, we revisit the spectral evolution of cool white dwarfs. Our
sample includes all spectroscopically identified white dwarfs cooler than 8300
K for which a parallax is available in Gaia DR2 and photometric observations
are available in Pan-STARRS1 and 2MASS. Except for a few cool carbon-polluted
objects, our models allow an excellent fit to the spectroscopic and photometric
observations of all objects included in our sample. We identify a decrease of
the ratio of hydrogen to helium-rich objects between 7500 K and 6250 K, which
we interpret as the signature of convective mixing. After this decrease,
hydrogen-rich objects become more abundant up to 5000 K. This puzzling
increase, reminiscent of the non-DA gap, has yet to be explained. At lower
temperatures, below 5000 K, hydrogen-rich white dwarfs become rarer, which
rules out the scenario according to which accretion of hydrogen from the
interstellar medium dominates the spectral evolution of cool white dwarfs.
