= Solution
The interface is a material surface, so its defining function $F=z-\zeta(x,y,t)$ obeys $DF/Dt=0$ on each side. Since $\mathbf u_i=\nabla\Phi_i$, the exact <kinematic boundary condition> is
$$
\partial_z\Phi_i=\partial_t\zeta+(\partial_x\Phi_i)\partial_x\zeta+(\partial_y\Phi_i)\partial_y\zeta\qquad(z=\zeta).
$$
Substitute $\Phi_i=U_ix+\varphi_i$. Products such as $\varphi_{ix}\zeta_x$ are second order, as is the correction from evaluating $\varphi_{iz}$ at $z=\zeta$ instead of zero. Linearization therefore gives
$$
\boxed{\partial_z\varphi_i=\partial_t\zeta+U_i\partial_x\zeta\qquad(z=0,\ i=1,2).}
$$
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