  Nucleon decays put extremely stringent bounds on baryon-number-violating
interactions. However, in case the corresponding operators involve only $\tau$
leptons, the direct two-body decays, e.g., $p\to\pi^0 \tau^+$, are
kinematically not allowed and nucleon decay can only proceed via an off-shell
$\tau$, leading to $p\to \pi^0\ell^+\nu_\ell\bar\nu_\tau$. To calculate such
processes, the momentum dependence of the form factors for nucleon-to-meson
transitions, which describe the hadronization of the underlying process at the
quark level, is needed. In this work, we point out new isospin and Fierz
relations among such proton-kaon matrix elements and calculate the momentum
dependence of the nucleon-to-meson form factors from the universal final-state
interactions in terms of pion-nucleon or kaon-nucleon scattering phase shifts.
We use these results to derive novel limits on the Wilson coefficients of the
baryon-number-violating dimension-$6$ operators involving a $\tau$ lepton,
which were previously unconstrained.
