I don't have time to give a proper answer now, but you may find it helpful to look at §8.1 of The formal theory of relative monads, which describes explicitly why weighted colimits in the usual sense (e.g. of Kelly) are captured by right lifts in $\mathrm{Dist}$. (The case of weighted limits is dual.)

Wood defines pointwise extensions in Abstract pro arrows I, as limits weighted by corepresentable distributors. If you look at the definition, it essentially coincides with the classical definition of pointwise right extension, e.g. here.

A pointwise right extension along $j \colon A \to B$ in $\mathrm{Cat}$ is precisely a limit weighted by the corepresentable $B(1, j)$, hence a right extension along $B(1, j)$ in $\mathrm{Dist}$. Is this what you are asking?

No rush, I was just curious :) I'm happy to wait until they're ready.

The concept of rank seems related.

Are you still planning to write up your notes about this result? I am interested in reading them.

@AlecRhea: I'm happy to assume as many universes as necessary to make everything work out. "Small" and "large" can be read as with respect to some universe, rather than literally sets and classes if that addresses any technical issues.

I think if there's an obstruction to covariant functoriality, this would already be a refutation of my question, because then taking presheaves on locally small categories would not have a freeness property. I was wondering whether this was the case earlier, but I didn't see how to show it.

One motivation for my question was discovering that the category of "petty presheaves" on a locally small category $C$ exhibited the "free well-cocompletion" of $C$ (Remark 4.39 of Lack–Tendas's Virtual concepts in the theory of accessible categories), which looks like a large cocompletion, and is in particular contained in the category of presheaves on $C$.

Yes, I wasn't sure how to word the title precisely but concisely. Is it easy to see why the answer to the question is no? (Kelly shows that one can characterise the colimits as those depending on diagrams, but I do not see how to show from this that a presentation purely by weights is impossible.)