  We study semigroups of convex monotone operators on spaces of continuous
functions and their behaviour with respect to $\Gamma$-convergence. In contrast
to the linear theory, the domain of the generator is, in general, not invariant
under the semigroup. To overcome this issue, we consider different versions of
invariant Lipschitz sets which turn out to be suitable domains for weaker
notions of the generator. The so-called $\Gamma$-generator is defined as the
time derivative with respect to $\Gamma$-convergence in the space of upper
semicontinuous functions. Under suitable assumptions, we show that the
$\Gamma$-generator uniquely characterizes the semigroup and is determined by
its evaluation at smooth functions. Furthermore, we provide Chernoff
approximation results for convex monotone semigroups and show that
approximation schemes based on the same infinitesimal behaviour lead to the
same semigroup. Our results are applied to semigroups related to stochastic
optimal control problems in finite and infinite-dimensional settings as well as
Wasserstein perturbations of transition semigroups.
