For a category $C$, there is a category Pro-$C$ whose objects are cofiltered diagrams $I \to C$ and whose morphisms are given by $$ {\rm Hom}(\{x_s\},\{y_t\}) = \varprojlim_t\ \varinjlim_s\ {\rm Hom}(x_s,y_t). $$ Generally, this category is fairly hard to work with. This is especially true because several types of maps of pro-objects are defined in terms of the existence of a representing map of diagrams with certain properties, and it can be very difficult to rectify several distinct properties at once.

One way to describe the category of pro-objects is by inverting morphisms. Specifically, we can form a more restricted category of pairs $(I,F)$ of a cofiltered index category $I$ and a diagram $F: I \to C$, with morphisms defined as pairs of a functor and a natural transformation of diagrams. Certain maps of cofiltered diagrams become isomorphisms of pro-objects (the most important ones being reindexing along a final subcategory). Inverting them gives us the pro-category.

In some sense, this automatically provides us with a "category with weak equivalences", but it's intrinsically very large and it's not necessarily clear if the "homotopy theory" is tractable.

Are there any circumstances under which the category of diagrams in $C$ automatically has the structure of a model category, with weak equivalences being pro-isomorphisms? In these cases, does the category Pro-$C$ have an interesting homotopy theory or are the mapping spaces essentially discrete?

Obviously being complete and cocomplete is going to be an obstacle to this kind of structure. Failing that, is there any further possibility of gaining control on the homotopy theory?

Having said all this, I've been a little bit vague about what I mean by a "map of diagrams" because I'd be open to the idea of having slightly restricted classes of maps in the definition.