  We measure the cross-correlation between galaxy weak lensing data from the
Kilo Degree Survey (KiDS-1000, DR4) and cosmic microwave background (CMB)
lensing data from the Atacama Cosmology Telescope (ACT, DR4) and the Planck
Legacy survey. We use two samples of source galaxies, selected with photometric
redshifts, $(0.1<z_{\rm B}<1.2)$ and $(1.2<z_{\rm B}<2)$, which produce a
combined detection significance of the CMB lensing/weak galaxy lensing
cross-spectrum of $7.7\sigma$. With the lower redshift galaxy sample, for which
the cross-correlation is detected at a significance of $5.3\sigma$, we present
joint cosmological constraints on the matter density parameter, $\Omega_{\rm
m}$, and the matter fluctuation amplitude parameter, $\sigma_8$, marginalising
over three nuisance parameters that model our uncertainty in the redshift and
shear calibration, and the intrinsic alignment of galaxies. We find our
measurement to be consistent with the best-fitting flat $\Lambda$CDM
cosmological models from both Planck and KiDS-1000. We demonstrate the capacity
of CMB-weak lensing cross-correlations to set constraints on either the
redshift or shear calibration, by analysing a previously unused high-redshift
KiDS galaxy sample $(1.2<z_{\rm B}<2)$, with the cross-correlation detected at
a significance of $7\sigma$. This analysis provides an independent assessment
for the accuracy of redshift measurements in a regime that is challenging to
calibrate directly owing to known incompleteness in spectroscopic surveys.
