A noncommutative domain Dfm(H), m∈N, of n-tuples of bounded linear operators on a Hilbert space H is associated with any positive regular free holomorphic function f. If Q⊂C〈Z1,…,Zn〉 is a set of polynomials in noncommutative indeterminates such that q(0)=0 for all Q∈Q, then the noncommutative varietyVf,Qm(H):={X=(X1,…,Xn)∈Dfm(H):q(X)=0 for all q∈Q} admits a universal modelB=(B1,…,Bn) acting on a model space Nf,Qm, which is a subspace of the full Fock space with n generators. If M is an arbitrary joint invariant subspace under the operators B1⊗IG,…,Bn⊗IG and the variety Vf,Qm(H) has the unique minimal dilation property, we provide an explicit description of the wandering subspaceW:=M⊖∑i=1n(Bi⊗IG)M and obtain a characterization of the inner multi-analytic operators Θ:Nf,Q1⊗G⁎→Nf,Qm⊗G, i.e. Θ|G⁎ is an isometry andΘ(G⁎)⊥(Bα⊗IG)Θ(G⁎),α∈Fn+,|α|≥1, where Fn+ is the free semigroup with n generators. We show that the noncommutative m-hyperball Dm(H) (which corresponds to f=Z1+⋯+Zn) has the unique minimal dilation property. When S=(S1,…,Sn) is the universal model for the unit ball D1(H) and W=(W1,…,Wn) is the universal model of the m-hyperball Dm(H), we provide a transfer function type characterization of the (S,W)-inner operators on Fock spaces. This extends to our noncommutative setting the corresponding results obtained by Olofsson (when n=1) and by Eschmeier in the multivariable commutative case.