The kinematic Sunyaev-Zeldovich (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 $sim 1$-$5times 10^{13}$ $h^{-1} M_{odot}$, 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.
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