  Based on the convergence of their infinitesimal generators in the mixed
topology, we provide a stability result for strongly continuous convex monotone
semigroups on spaces of continuous functions. In contrast to previous results,
we do not rely on the theory of viscosity solutions but use a recent comparison
principle which uniquely determines the semigroup via its $\Gamma$-generator
defined on the Lipschitz set and therefore resembles the classical analogue
from the linear case. The framework also allows for discretizations both in
time and space and covers a variety of applications. This includes Euler
schemes and Yosida-type approximations for upper envelopes of families of
linear semigroups, stability results and finite-difference schemes for convex
HJB equations, Freidlin-Wentzell-type results and Markov chain approximations
for a class of stochastic optimal control problems and continuous-time Markov
processes with uncertain transition probabilities.
