The comprehensive treatment can be found in section 2.4 of:
Schiff, Joel L. Normal families. Universitext. Springer-Verlag, New York, 1993. xii+236 pp. ISBN: 0-387-97967-0

Here is what he calls Vitali-Porter theorem: Let $\{f_n\}$ be a locally bounded sequence of analytic functions in a domain $\Omega$ such that the limit $\lim_{n\to \infty}f_n(z)$ exists for all $z$ belonging to a subset $E \subset \Omega$ which has an accumulation point in $\Omega$. Then $f_n$ converges uniformly on compact subsets of $\Omega$ to an analytic function.

In this text it is explicitly shown that this theorem is equivalent to Montel's theorem on bounded families of analytic functions. Other texts (e.g. Goluzin) usually only deduce Vitali's theorem from Montel's theorem, if they treat it at all. Also, several generalizations are mentioned at the end of the section. All seem to concern sequences rather than general nets, but since in Montel's theorem the family is not required to be countable, perhaps you can somehow get what you want from these results.