  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., $u\sim \Delta$. 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).
