  The kinematics of the Milky Way disc as a function of age are well measured
at the solar radius, but have not been studied over a wider range of
Galactocentric radii. Here, we measure the kinematics of mono-age,
mono-$\mathrm{[Fe/H]}$ populations in the low and high $\mathrm{[\alpha/Fe]}$
discs between $4 \lesssim R \lesssim 13$ kpc and $|z| \lesssim 2$ kpc using
65,719 stars in common between APOGEE DR14 and $\it{Gaia}$ DR2 for which we
estimate ages using a Bayesian neural network model trained on asteroseismic
ages. We determine the vertical and radial velocity dispersions, finding that
the low and high $\mathrm{[\alpha/Fe]}$ discs display markedly different
age--velocity-dispersion relations (AVRs) and shapes $\sigma_z/\sigma_R$. The
high $\mathrm{[\alpha/Fe]}$ disc has roughly flat AVRs and constant
$\sigma_z/\sigma_R = 0.64\pm 0.04$, whereas the low $\mathrm{[\alpha/Fe]}$ disc
has large variations in this ratio which positively correlate with the mean
orbital radius of the population at fixed age. The high $\mathrm{[\alpha/Fe]}$
disc component's flat AVRs and constant $\sigma_z/\sigma_R$ clearly indicates
an entirely different heating history. Outer disc populations also have flatter
radial AVRs than those in the inner disc, likely due to the waning effect of
spiral arms. Our detailed measurements of AVRs and $\sigma_z/\sigma_R$ across
the disc indicate that low $\mathrm{[\alpha/Fe]}$, inner disc ($R \lesssim
10\,\mathrm{kpc}$) stellar populations are likely dynamically heated by both
giant molecular clouds and spiral arms, while the observed trends for outer
disc populations require a significant contribution from another heating
mechanism such as satellite perturbations. We also find that outer disc
populations have slightly positive mean vertical and radial velocities, likely
because they are part of the warped disc.
