Due to the similarity of L'Hôpital's rule and Stolz-Cesàro theorem I guess that it must exists a generalized theorem where each of these two theorems are special cases.
Note that in the setting of finite calculus the $\Delta $ operator is the inverse of the $\sum$ operator, that is, if $f:\Bbb N \to \Bbb R $ then $\Delta f$ can be seen as the finite calculus derivative of $f$ and $\sum_0^n f$ as a finite calculus primitive of $f$.
Indeed in a measure-theoretic context one can see $f$ as the Radon-Nikodym derivative of $f \,\mathrm d \mu $ where $\mu $ is the counting measure in $\Bbb N $ (and $\Delta f$ can be seen as some Radon-Nikodym derivative of the measure $\Delta f \,\mathrm d \mu $ respect to the counting measure $\mu $).
So suppose we have an operator $T$, what condition we will need on $T$ (or the space where it is defined) to have a version of the Stolz-Cesàro theorem? I mean, when it holds that $$ \lim_{n\to \infty }\frac{Tf_n}{Tg_n}=h \implies \lim_{n\to \infty }\frac{f_n}{g_n}=h $$ assuming that $(g_n)$ is eventually not zero or that it diverges to infinity, and that the RHS of above is an indeterminate form of the kind $0/0$ or $\infty/\infty$? There is some theorem about that, or some paper or bibliography?