Let D denote the unit disc of C and let Ω denote the unit ball B_n of Cⁿ or the unit polydisc Dⁿ, n≥ 2. Given a non-constant holomorphic function b: Ω → D, we study the corresponding family σ_α[ b], α∈ ∂D, of Clark measures on ∂Ω. For Ω = B_n and an inner function I: B_n→ D, we show that the property σ₁[ I] ≪ σ₁[ b] is directly related to the membership of an appropriate function in the de Branges–Rovnyak space H(b).