  Teukolsky equations for $|s|=2$ provide efficient ways to solve for curvature
perturbations around Kerr black holes. Imposing regularity conditions on these
perturbations on the future (past) horizon corresponds to imposing an in-going
(out-going) wave boundary condition. For exotic compact objects (ECOs) with
external Kerr spacetime, however, it is not yet clear how to physically impose
boundary conditions for curvature perturbations on their boundaries. We address
this problem using the Membrane Paradigm, by considering a family of fiducial
observers (FIDOs) that float right above the horizon of a linearly perturbed
Kerr black hole. From the reference frame of these observers, the ECO will
experience tidal perturbations due to in-going gravitational waves, respond to
these waves, and generate out-going waves. As it also turns out, if both
in-going and out-going waves exist near the horizon, the Newman Penrose (NP)
quantity $\psi_0$ will be numerically dominated by the in-going wave, while the
NP quantity $\psi_4$ will be dominated by the out-going wave. In this way, we
obtain the ECO boundary condition in the form of a relation between $\psi_0$
and the complex conjugate of $\psi_4$, in a way that is determined by the ECO's
tidal response in the FIDO frame. We explore several ways to modify
gravitational-wave dispersion in the FIDO frame, and deduce the corresponding
ECO boundary condition for Teukolsky functions. We subsequently obtain the
boundary condition for $\psi_4$ alone, as well as for the Sasaki-Nakamura and
Detweiler's functions. As it also turns out, reflection of spinning ECOs will
generically mix between different $\ell$ components of the perturbations
fields, and be different for perturbations with different parities. We also
apply our boundary condition to computing gravitational-wave echoes from
spinning ECOs, and solve for the spinning ECOs' quasi-normal modes.
