The kinematic Sunyaev-Zel'dovich (kSZ) effect -- the Doppler boosting of
cosmic microwave background (CMB) photons scattering off free electrons with
non-zero line-of-sight velocity -- is an excellent probe of the distribution of
baryons in the Universe. In this paper, we measure the kSZ effect due to
ionized gas traced by infrared-selected galaxies from the \emph{unWISE}
catalog. We employ the "projected-field" kSZ estimator, which does not require
spectroscopic galaxy redshifts. To suppress non-kSZ foreground signals
associated with the galaxies (e.g., dust emission and thermal SZ), this
estimator requires cleaned CMB maps, which we obtain from \emph{Planck} and
\emph{WMAP} data. Using a new "asymmetric" estimator that combines different
foreground-cleaned CMB maps to maximize the signal-to-noise, we measure the
kSZ$^2$-galaxy cross-power spectrum for three subsamples of the \emph{unWISE}
galaxy catalog, which peak at mean redshifts $z \approx$ 0.6, 1.1, and 1.5,
have average halo mass $\sim 1$-$5\times 10^{13}$ $h^{-1} M_{\odot}$, and in
total contain over 500 million galaxies. After marginalizing over CMB lensing
contributions, we measure the amplitude of the kSZ signal $A_{\rm kSZ^2} = 0.42
\pm 0.31(stat.) \pm 0.14(sys.)$, $5.02 \pm 1.01(stat.) \pm 0.47(sys.)$, and
$8.23 \pm 3.23(stat.) \pm 1.60(sys.)$, for the three subsamples, where $A_{\rm
kSZ^2} = 1$ corresponds to our fiducial model. The combined kSZ detection S/N
$>$ 5. We discuss possible explanations for the excess kSZ signal associated
with the $z \approx 1.1$ sample, and show that foreground contamination in the
CMB maps is very unlikely to be the cause. Our measurements illustrate clearly
that no baryons are missing on large scales at low redshifts.
