Starting from the general Lippmann-Schwinger equation, we study the effect of inelastic processes on the scattering of three-body states. We show that it is possible to incorporate the inelastic effects into Faddeev-type equations in two different ways. The first approach is a straightforward generalization of the single-channel Faddeev equations to the multichannel form. However, we have to introduce the concepts of position and particle labeling in order to obtain a meaningful generalization. The second approach, which is derived from an extension of the concept of a complex potential, yields single-channel Faddeev equations with a modified input. The essential difference between this input and the input for the elastic case is the presence of a completely connected term. The two approaches are shown to be equivalent in their common region of validity. Under the resonance approximation, both approaches yield one-dimensional equations. The structure of the completely connected term is investigated for certain specific models and further simplifications on it are obtained. We also observe that the concept of the inelasticity parameter in the two-body case does not seem to have a natural generalization.