Summary: Let \(\widetilde{S}^t_A(\mathbf{B}^n)\) be the family of normalized univalent mappings on the Euclidean unit ball \(\mathbf{B}^n\) in \(\mathbf{C}^n\), which have generalized parametric representation with respect to time-dependent operators \(A\in\widetilde{\mathcal{A}}\), where \(\widetilde{\mathcal{A}}\) is a family of measurable mappings from \([0,\infty)\) into \(L(\mathbf{C}^n)\) with some particular properties. Also, let \(\widetilde{\mathcal{R}}_T(\mathrm{id}_{\mathbf{B}^n},(\mathcal{N}_{A(t)})_{t\in[T_0,T]})\) be the time-\(T\)-reachable family of the control system \(\mathcal{C}([T_0,T],(\mathcal{N}_{A(t)})_{t\in[T_0,T]})\), where \(A\in\widetilde{\mathcal{A}}\) and \(T_0\geq0\). In this paper we obtain certain convergence results for the families \(\widetilde{S}^t_A(\mathbf{B}^n)\) and \(\widetilde{\mathcal{R}}_T(\mathrm{id}_{\mathbf{B}^n},(\mathcal{N}_{A(t)})_{t\in[T_0,T]})\) with respect to the Hausdorff metric \(\rho\) on \(H(\mathbf{B}^n)\). These results may be seen as dominated convergence type theorems for time-dependent operators \(A\in\widetilde{\mathcal{A}}\). In particular, we obtain related convergence results for the family \(S^0_{\mathbf{A}}(\mathbf{B}^n)\) (resp. for the family \(\widehat{S}_{\mathbf{A}}(\mathbf{B}^n)\)) of mappings with \(\mathbf{A}\)-parametric representation on \(\mathbf{B}^n\) (resp. of spirallike mappings on \(\mathbf{B}^n\) with respect to \(\mathbf{A}\)), in the case that \(\mathbf{A}\in L(\mathbf{C}^n)\) is a linear operator with \(k_+(\mathbf{A})0\). Finally, we obtain some sufficient conditions related to \(A\in\widetilde{\mathcal{A}}\), which yield the equality \(\widetilde{S}^t_A(\mathbf{B}^n)=S^0(\mathbf{B}^n)\), for all \(t\geq0\), where \(S^0(\mathbf{B}^n)\) is the family of normalized univalent mappings with usual parametric representation on \(\mathbf{B}^n\).