Let B_d denote the unit ball of C^d, d≥1. Given a holomorphic function φ:B_d→B₁, we study the corresponding family σ_α[φ], α∈∂B₁, of Clark measures on the unit sphere ∂B_d. If φ is an inner function, then we introduce and investigate related unitary operators U_α mapping analogs of model spaces onto L²(σ_α), α∈∂B₁. In particular, we explicitly characterize the set of U_α ^⁎f such that fσ_α is a pluriharmonic measure. Also, for an arbitrary holomorphic φ:B_d→B₁, we use the family σ_α[φ] to compute the essential norm of the composition operator C_φ:H²(B₁)→H²(B_d).