The effect of an environment on ultrafast predissociation processes is modeled in terms of a Markovian, Lindblad-type master equation in the coordinate representation. The analysis focuses on the effects of vibrational and electronic dephasing on one hand and "indirect", environment-induced electronic transitions on the other hand. The latter not only exert a quenching effect on coherent curve-crossing dynamics, as shown for a system exhibiting periodic Landau-Zener-type crossings, but may largely determine the dynamics if the system features no intrinsic electronic coupling. This provides a simplified description, restricted to the Markovian limit, for purely environment-induced processes such as the predissociation of I-2 in its BOu+ state. Numerical integration of the coordinate-space master equation is performed by a split-propagator technique. The problem of defining absorbing boundaries is solved by introducing an emission process, localized around the grid boundary, to an auxiliary state that is otherwise not involved in the dynamics.