  Let $B_d$ denote the unit ball of $\mathbb{C}^d$, $d\ge 1$. Given a
holomorphic function $\varphi: B_d \to B_1$, we study the corresponding family
$\sigma_\alpha[\varphi]$, $\alpha\in\partial B_1$, of Clark measures on the
unit sphere $\partial B_d$. If $\varphi$ is an inner function, then we
introduce and investigate related unitary operators $U_\alpha$ mapping analogs
of model spaces onto $L^2(\sigma_\alpha)$, $\alpha\in\partial B_1$. In
particular, we explicitly characterize the set of $U_\alpha^* f$ such that
$f\sigma_\alpha$ is a pluriharmonic measure. Also, for an arbitrary holomorphic
$\varphi: B_d \to B_1$, we use the family $\sigma_\alpha[\varphi]$ to compute
the essential norm of the composition operator $C_\varphi: H^2(B_1)\to
H^2(B_d)$.
