  Measurements of secondary cosmic microwave background (CMB) anisotropies,
such as the Sunyaev-Zel'dovich (SZ) effect, will enable new tests of neutrino
and dark sector properties. The kinetic SZ (kSZ) effect is produced by
cosmological flows, probing structure growth. Ultra-light axions (ULAs) are a
well-motivated dark-matter candidate. Here the impact of ULA dark matter (with
mass $10^{-27}~{\rm eV}$ to $10^{-23}~{\rm eV}$) on kSZ observables is
determined, applying new analytic expressions for pairwise cluster velocities
and Ostriker-Vishniac signatures in structure-suppressing models. For the
future CMB-S4 and ongoing DESI galaxy surveys, the kSZ effect (along with
primary anisotropies) will probe ULA fractions $\eta_a =
\Omega_{\rm{axion}}/\Omega_{\rm DM}$ as low as $\sim 5\%$ if $m_{a}\simeq
10^{-27}~{\rm eV}$ (at 95\% C.L.), with sensitivity extending up to
$m_{a}\simeq 10^{-25}~{\rm eV}$. If reionization and the primary CMB can be
adequately modeled, Ostriker-Vishniac measurements could probe values
$\eta_{a}\simeq 10^{-3}$ if $10^{-27}~{\rm eV}\lesssim m_{a}\lesssim
10^{-24}~{\rm eV}$, or $\eta_{a}\simeq 1$ if $m_{a}\simeq 10^{-22}~{\rm eV}$,
within the fuzzy dark matter window.
