  We critically examine the ability of future neutrino telescopes, including
Baikal-GVD, KM3NeT, P-ONE, TAMBO, and IceCube-Gen2, to determine the flavor
composition of high-energy astrophysical neutrinos, ie, the relative number of
$\nu_e$, $\nu_\mu$, and $\nu_\tau$, in light of improving measurements of the
neutrino mixing parameters. Starting in 2020, we show how measurements by JUNO,
DUNE, and Hyper-Kamiokande will affect our ability to determine the regions of
flavor composition at Earth that are allowed by neutrino oscillations under
different assumptions of the flavor composition that is emitted by the
astrophysical sources. From 2020 to 2040, the error on inferring the flavor
composition at the source will improve from $> 40\%$ to less than $6\%$. By
2040, under the assumption that pion decay is the principal production
mechanism of high-energy astrophysical neutrinos, a sub-dominant mechanism
could be constrained to contribute less than 20\% of the flux at 99.7\%
credibility. These conclusions are robust in the nonstandard scenario where
neutrino mixing is non-unitary, a scenario that is the target of
next-generation experiments, in particular the IceCube-Upgrade. Finally, to
illustrate the improvement in using flavor composition to test
beyond-the-Standard-Model physics, we examine the possibility of neutrino decay
and find that, by 2040, combined neutrino telescope measurements will be able
to limit the decay rate of the heavier neutrinos to below $1.8\times 10^{-5}
(m/\mathrm{eV})$~s$^{-1}$, at 95\% credibility.
