Let $(A,\varphi)$ be a noncommutative probability space over $\mathbb{C}$, let $I$ be an index set, and let $(A_i)_{i \in I}$ be a freely independent family of unital subalgebras of $A$. For each $m \in \mathbb{N}$, let $\kappa_m: A^m \to \mathbb{C}$ denote the $m$-th free cumulant functional associated to $\varphi$ through the free moment-cumulant formula over noncrossing partitions, and assume the mixed-cumulant characterization of freeness holds for this family: whenever $m \in \mathbb{N}$, $j_1,\dots,j_m \in I$, $a_\ell \in A_{j_\ell}$ for $1 \leq \ell \leq m$, and the indices $j_1,\dots,j_m$ are not all equal, one has $\kappa_m(a_1,\dots,a_m)=0$. Suppose that, for every $i \in I$ and every $m \in \mathbb{N}$, the restricted multilinear functional $\kappa_m|_{A_i^m}: A_i^m \to \mathbb{C}$ is known. Then for every $n \in \mathbb{N}$, every function $c: \{1,\dots,n\} \to I$, and every tuple $(x_1,\dots,x_n) \in A_{c(1)} \times \cdots \times A_{c(n)}$, the mixed moment $\varphi(x_1 \cdots x_n)$ is uniquely determined by the family of restricted cumulant functionals $\{\kappa_m|_{A_i^m}: i \in I, m \in \mathbb{N}\}$.