We investigate field-induced transformations in the dynamical response of the XXZ model on the triangular lattice that are associated with the anharmonic magnon coupling and decay phenomena. Detailed theoretical predictions are made for Ba3CoSb2O9, which provides a close realization of the spin-1/2 XXZ model. We demonstrate that dramatic modifications in the magnon spectrum must occur in low out-of-plane fields that are easily achievable for this material. The hallmark of the effect is a coexistence of the clearly distinct well-defined magnon excitations with significantly broadened ones in different regions of the k-omega space. The field-induced decays are generic for this class of models and become more prominent at larger anisotropies and in higher fields.