Let \(X_1,X_2, \dots, X_n\) be metric spaces, \(\text{CL} (X_i)\) be the set of nonempty closed subsets of \(X_i\), \(i=1, \dots, n\), and \(H_i\) be the related Hausdorff metric. The authors, inspired by a result of \textit{H. Kaneko} and the reviewer [Int. J. Math. Math. Sci. 12, No. 2, 257-262 (1989; Zbl 0671.54023)] consider two systems of maps \(\{T_1, \dots, T_n\}\) and \(\{P_1, \dots, P_n\}\), being \(T_i\): \(X\to X_i\) and \(P_i\): \(X\to \text{CL} (X_i)\) with \(X= X_1 \times \cdots \times X_n\). They are called asymptotically commuting if \[ H_i \bigl(P_i (T_1 x^m, \dots, T_nx^m),\;T_i(P_1x^m, \dots, P_nx^m) \bigr) \to 0 \] whenever \(\{x^m\}\) is a sequence in \(X\) such that \(T_ix^m \to x_i\in M_i\) and \(P_ix^m \to M_i\in \text{CL} (x_i)\). Then the authors obtain coincidence point theorems for such maps which satisfy additionally conditions similar to those for hybrid contractions. Many theorems of other authors are then generalized.