We investigate the effects of dissipation on the quantum dynamics of many-body systems at quantum transitions, especially considering those of the first order. This issue is studied within the paradigmatic one-dimensional quantum Ising model. We analyze the out-of-equilibrium dynamics arising from quenches of the Hamiltonian parameters and dissipative mechanisms modeled by a Lindblad master equation, with either local or global spin operators acting as dissipative operators. Analogously to what happens at continuous quantum transitions, we observe a regime where the system develops a nontrivial dynamic scaling behavior, which is realized when the dissipation parameter $u$ (globally controlling the decay rate of the dissipation within the Lindblad framework) scales as the energy difference $Delta$ of the lowest levels of the Hamiltonian, i.e., $usim Delta$. However, unlike continuous quantum transitions where $Delta$ is power-law suppressed, at first-order quantum transitions $Delta$ is exponentially suppressed with increasing the system size (provided the boundary conditions do not favor any particular phase).
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