  Typically, constraints on parameters of the effective field theory (EFT) of
dark energy have been obtained in the Jordan frame, where matter fields are
minimally coupled to gravity. To connect these constraints with those of the
EFT of black hole perturbations with a timelike scalar profile, it is necessary
to perform a frame transformation on the EFT in general. In this paper, we
study the conformal/disformal transformation of EFT parameters on an arbitrary
background. Furthermore, we explore the effect of an EFT operator $M_6(r)
\bar{\sigma}^{\mu}_{\nu} \delta K^{\nu}_{\alpha} \delta K^{\alpha}_{\mu}$,
which is elusive to the LIGO/Virgo bound on gravitational-wave speed, on the
dynamics of odd-parity black hole perturbations. Intriguingly, a deviation from
luminal propagation shows up only in the vicinity of the black hole, and the
speeds of perturbations in the radial and angular directions are different in
general due to the traceless part $\bar{\sigma}^\mu_\nu$ of the background
extrinsic curvature. This study establishes an important link between
cosmological constraints and those obtained in the black hole regime.
