  The fine-structure constant, $\alpha$, controls the strength of the
electromagnetic interaction. There are extensions of the standard model in
which $\alpha$ is dynamical on cosmological length and time-scales. The physics
of the cosmic microwave background (CMB) depends on the value of $\alpha$. The
effects of spatial variation in $\alpha$ on the CMB are similar to those
produced by weak lensing: smoothing of the power spectrum, and generation of
non-Gaussian features. These would induce a bias to estimates of the
weak-lensing potential power spectrum of the CMB. Using this effect, Planck
measurements of the temperature and polarization power spectrum, as well as
estimates of CMB lensing, are used to place limits (95% C. L.) on the amplitude
of a scale-invariant angular power spectrum of $\alpha$ fluctuations relative
to the mean value ($C_L^{\alpha}=A^\alpha_{\rm SI}/[L(L+1)]$) of $A^\alpha_{\rm
SI}\leq 1.6 \times 10^{-5}$. The limits depend on the assumed shape of the
$\alpha$-fluctuation power spectrum. For example, for a white noise angular
power spectrum ($C_L^{\alpha}=A^\alpha_{\rm WN}$), the limit is $A^\alpha_{\rm
WN}\leq 2.3 \times 10^{-8}$. It is found that the response of the CMB to
$\alpha$ fluctuations depends on a separate-universe approximation, such that
theoretical predictions are only reliable for $\alpha$ multipoles with
$L\lesssim 100$ . An optimal trispectrum estimator can be constructed and it is
found that it is only marginally more sensitive than lensing techniques for
Planck but significantly more sensitive when considering the next generation of
experiments. For a future CMB experiment with cosmic-variance limited
polarization sensitivity (e.g., CMB-S4), the optimal estimator could detect
$\alpha$ fluctuations with $A^\alpha_{\rm SI}>1.9 \times 10^{-6}$ and
$A^\alpha_{\rm WN} > 1.4 \times 10^{-9}$.
