I would like to have an explicit description of the left/right Kan lift of a functor $F$ through $G$, $\text{Lift}_GF$/$\text{Rift}_GF$ in terms of coends/ends (this can be done for Kan extensions, so I hardly believe there is no way to dualize the argument).
Google is of little help, a part from a couple of hints on how to define $\text{Rift}$ of a profunctor through another.
Can you help me? Feel free to suppose the completeness/cocompleteness you need to define the right end/coend.
Such a characterisation is not possible in $\mathbf{Cat}$ for trivial reason --- the dual of a functor generally cannot be thought of as a functor. In an analogical situation in $\mathbf{Set}$, for exactly the same reason, we do not expect that every colimit can be expressed in a canonical way as a limit.
For example, let us write $|2|$ for the discrete category consisting of two objects: $0$ and $1$, and let $\delta_0 \colon \mathbb{C} \rightarrow |2|$, $\delta_1 \colon \mathbb{D} \rightarrow |2|$ be the obvious constant functors (with disjoint images) from non-trivial categories $\mathbb{C}, \mathbb{D}$. Then there are no liftings of $\delta_0$ through $\delta_1$, no matter how complete and cocomplete $\mathbb{C}$ and $\mathbb{D}$ are. In fact, for any functor $F \colon \mathbb{C} \rightarrow \mathbb{D}$, there are no natural transformations $\delta_1 \circ F \rightarrow \delta_0$ nor $\delta_0 \rightarrow \delta_1 \circ F$. The same is true, if we substitute category $|2|$ with its free cocompletion $\mathbf{Set}^{|2|^{op}}$.
Notice also, that the formula for Kan liftings of profunctors, defines liftings internal to the bicategory of profunctors, and not internal to the 2-category of categories.
Let me recall that a Kan lifting in $\mathbf{Cat}$ is defined as a Kan extension in $\mathbf{Cat}^{op}$. The deep reason, why we do not have nice formulae for Kan liftings, is that $\mathbf{Cat}^{op}$, from the perspective of our covariant world, is not as nice as $\mathbf{Cat}$. Specifically, there is no way to define the concept of (co)ends in $\mathbf{Cat}^{op}$, because the internal calculi of $\mathbf{Cat}^{op}$ is not expressible enough ($\mathbf{Cat}^{op}$ does not have the canonical Yoneda structure).
Saying that, it is good to know, that in most situations we care about absolute Kan liftings, which may be expressed in the usual hom-set manner (these formulae can be further generalised inside any 2-category):
$$\hom(F(-), G(=)) \approx \hom(-, \mathit{Rift}_F(G)(=))$$
and:
$$\hom(F(-), G(=)) \approx \hom(\mathit{Lift}_G(F)(-), =)$$
