How to characterize the net convergence in final topology?

There is no simple characterization of when a net converges in the final topology. In particular, arguably the simplest case is when your family just consists of a single space $Y$ with a surjective map $f:Y\to X$. In that case the final topology is just the quotient topology, but there is no simple description of when a net converges in the quotient topology in terms of convergence of nets in the original space. For examples of how some naive guesses can go wrong, you may be interested in the post Lifting a convergent net through a quotient map.

To give a bit of a broader perspective, convergence of a net in a space $X$ is equivalent to continuity of a certain map $I\to X$ for a certain space $I$ (see this answer of mine). The initial topology is a type of limit in the category of topological spaces, and maps into a limit are characterized by a universal property, and so convergence of nets in a limit space has a simple characterization. On the other hand, the final topology is a colimit, and there is no universal property for maps into a colimit (instead the universal property is for maps out of it). So it should not be surprising that there is no nice characterization of convergence of nets in a colimit.