Premises:
definition (class): A primitive object. Left undefined.
definition (proper class and set): A class is proper iff it is not a member of another class. A class which is a member of another class is called a set.
definition (family of sets): A class $\mathscr F$ is a family of sets on a class $C$ iff $\bigcup \mathscr F=C$. (typically, this would be called a "cover," but every family of sets is a cover of its union anyway, so the distinction is largely irrelevant; and "family" just sounds better than "cover".)
We call a family $\mathscr F$ a Russell family$^1$ iff $\bigcup \mathscr F\in\mathscr F$. Many important families of sets are Russell families. For exmple, all of the following are Russell families
Powersets
Topologies
Filters
Join-complete lattices and semilattices
$\lambda$-systems and $\sigma$-algebras
Problem:
No large family of sets (a family of sets over a proper class) can be a Russell family. The proof is straightforward:
Suppose towards contradiction that $\mathscr F$ is a Russell family on a proper class $C$. Since $C$ is a proper class, $C\notin X$ for any class $X$. In particular, $C\notin\mathscr F$. But $\mathscr F$ is a Russel family, so $C=\bigcup\mathscr F\in\mathscr F$ - a contradiction.
Unfortunately, there are times when we would like to talk about conventionally Russell families on proper classes. The obvious example for me is the order topology on the class of ordinals. By the previous, there can't really be any topology on the class of ordinals - or any other proper class, for that matter. Similarly, we cannot rightly discuss ultrafilters on proper classes, or the lattice of sets formed by inclusion on a proper class, because all of these structures are Russell families.
This does not stop set theorists from doing so anyway.
Now I can see a few different ways to clean things up: we can avoid talking about large families altogether and focus on the properties of arbitrary families on subsets of proper classes instead. e.g. "fix an ordinal $\gamma$..."; we can redefine various structures so that they are only Russell families when the domain is a set, or so that they don't need to be Russell families in the first place; we can introduce a heirarchy of "logical types" a la Principia Mathematica - with sets as 1-types, classes of sets which are not themeselves sets as 2-types, collections of classes as 3-types, and so on; or we can don robes and chant "higher category theory" until all the boomers leave the room.
Whatever the case, is there a widely accepted convention for handling large families of sets?
$^1$ this is just a placeholder name; feel free to change it if there's already canonical name for this property.