The generalized string-flip model for quantum scattering in a system of two N-body clusters is under consideration. The extra channel interpreted as the quark compound bag is considered which generates energy-dependent integral boundary conditions in the effective boundary problem. It is shown that the standard string-flip model can be generalized for N identical particles in a cluster. The effective Hamiltonian and effective integral equations for the partial s-wave S matrix are proven to be form invariant with respect to N. The effective configuration space is shown to be two-dimensional for every N and its geometry for different N is discussed. Results of the numerical calculations of s-wave scattering phases, inelasticity parameters, and scattering lengths for N=2 (meson-meson scattering) and N=3 (baryon-baryon scattering) are presented. The resonance influence of the extra channel is investigated. Generalizations including spin-spin interaction and quark color group SU(n) are also presented.