  Asymmetric emission of gravitational waves during a compact binary
coalescence results in the loss of linear momentum and a corresponding `kick'
or recoil on the binary's center of mass. This leads to a direction-dependent
Doppler shift of the ringdown gravitational waveform. We quantify the
measurability of the kick imparted to the remnant black hole in a binary black
hole merger. Future ground- and space-based gravitational wave detectors will
measure this effect to within $\sim 2\%$ to $\sim 30\%$ for a subset of their
expected observed sources. Certain binary configurations in the LISA band may
allow a sub-percent-level measurement of this effect. This direct measurement
of black hole kicks can also facilitate a novel test of general relativity
based on linear momentum balance. We formulate this kick consistency test via
measurement of a null variable that quantifies the difference between the
inferred kick (using numerical relativity) and that observed via the
Doppler-shifted ringdown signal. This null variable can be constrained (at 90%
confidence) to $\sim 10\%$ to $30\%$ with Cosmic Explorer and to $\sim 3\%$ to
$12\%$ with LISA.
