  We show stability and locality of the minimal supersolution of a forward
backward stochastic differential equation with respect to the underlying
forward process under weak assumptions on the generator. The forward process
appears both in the generator and the terminal condition. Painlev\'e-Kuratowski
and Convex Epi-convergence are used to establish the stability. For Markovian
forward processes the minimal supersolution is shown to have the Markov
property. Furthermore, it is related to a time-shifted problem and identified
as the unique minimal viscosity supersolution of a corresponding PDE.
