Two alternative versions of practicable connected kernel theories of nuclear reactions are proposed. The basic assumption is that the state of a many-body system can be approximated by a superposition of two- and three-cluster states corresponding to various possible reaction processes. The approach is based on the concept of transitions due to particle exchange as in the AmadoLovelace (AL) formalism. All important three-cluster partitions can be incorporated via multichannel couplings in two-cluster subsystems. The simpler of two models, which is called the multi three-cluster coupling (MTCC) model, is a direct extension of the AL formalism. Using the separable representation of two-cluster potentials, the AL type coupled equations among reaction amplitudes are postulated. The MTCC equation is shown to be reduced to the equation with the same structure as the Faddeev equation. Thus, while the MTCC model can treat a much wider class of nuclear reactions than the Faddeev approach, the calculation is no more complicated than that. The basic assumption in this model, as well as in the AL and AGS theories, is shown to contain some degree of inconsistency regarding the treatment of bound state pole parts of interacting pairs. This is remedied in the other model, which we call the multi two- and three-cluster coupling (MTTC) model. This is the most extended MTCC model that is possible within the two- and three-cluster coupling model. In the MTTC treatment, it is shown that sequential (multi-step) transfers can also be accommodated in addition to all various processes that are possible in the MTCC approach.