  We aim to map the distribution and kinematics of molecular and ionized gas in
a sample of active galaxies, to quantify the nuclear inflows and outflows.
Here, we analyze the nuclear kinematics of NGC 1566 via ALMA observations of
the CO J:2-1 emission at 24 pc spatial and $\sim$2.6 km s$^{-1}$ spectral
resolution, and Gemini-GMOS/IFU observations of ionized gas emission lines and
stellar absorption lines at similar spatial resolution, and 123 km s$^{-1}$ of
intrinsic spectral resolution. The morphology and kinematics of stellar,
molecular (CO) and ionized ([N II]) emission lines are compared to the
expectations from rotation, outflows, and streaming inflows. While both ionized
and molecular gas show rotation signatures, there are significant non-circular
motions in the innermost 200 pc and along spiral arms in the central kpc (CO).
The nucleus shows a double-peaked CO profile (Full Width at Zero Intensity of
200 km s$^{-1}$), and prominent ($\sim$80 km s$^{-1}$) blue and redshifted
lobes are found along the minor axis in the inner arcseconds. Perturbations by
the large-scale bar can qualitatively explain all features in the observed
velocity field. We thus favour the presence of a molecular outflow in the disk
with true velocities of $\sim$180 km s$^{-1}$ in the nucleus and decelerating
to 0 by $\sim$72 pc. The implied molecular outflow rate is
$5.6~[M_{o}yr^{-1}]$, with this gas accumulating in the nuclear 2 arcsec arms.
The ionized gas kinematics support an interpretation of a similar, but more
spherical, outflow in the inner 100 pc, with no signs of deceleration. There is
some evidence of streaming inflows of $\sim$50 km s$^{-1}$ along specific
spiral arms, and the estimated molecular mass inflow rate,
$\sim0.1~[M_{o}yr^{-1}]$, is significantly larger than the SMBH accretion rate
($\dot{m}=4.8\times10^{-5}~[M_{o}yr^{-1}]$).
