In a CFT one can expand correlation functions in conformal blocks, for example the four-point function can be written (schematically) as:

$$\langle \mathcal{O}_1 (x_1) \mathcal{O}_2 (x_2) \mathcal{O}_3 (x_3) \mathcal{O}_4 (x_4) \rangle \sim \sum_k \lambda_{12k} \lambda_{34k}\, g_k(u,v)\,, \tag{1}$$

with $$u,v$$ the independent anharmonic cross-ratios and $$\lambda_{ijk}$$ the OPE coefficients of three-point functions $$\langle \mathcal{O}_i \mathcal{O}_j \mathcal{O}_k \rangle$$.

My question is simple: this expansion is a non-perturbative statement based on (conformal) symmetry only, so if the AdS/CFT claim is right it should also be valid for the dual theories. However so far I have never seen people deriving such an expansion from the gravity side. Have some people actually managed to do so? If not, why can we not extract this expansion from the symmetries of the gravity side?

There are many papers which explore the conformal block expansion from the gravity side. A consistency check with the OPE was one of the motivations for https://arxiv.org/abs/hep-th/9903196 which carried out the first calculation of a 4-point function in $$\mathcal{N} = 4$$ Super Yang-Mills. A more recent example is https://arxiv.org/abs/1803.01379 which solves for some of the exchanged scaling dimensions and OPE coefficients in ABJM theory. There are also expansions in https://arxiv.org/abs/1508.00501 which show what an abstract Witten diagram looks like in terms of conformal blocks without committing to any particular bulk theory.
All of these checks have to contend with the fact that perturbative calculations can only shift scaling dimensions by an infinitesimal amount. For interacting theories, we expect to see many blocks $$g_{\Delta, \ell}(u, v)$$ where $$\Delta$$ is not an integer. However, the closest we can get in AdS / CFT is something like \begin{align} g_{n + \frac{1}{c} \gamma + \dots, \ell}(u, v) = g_{n, \ell}(u, v) + \frac{1}{c} \gamma \partial_n g_{n,\ell}(u, v) + O\left ( \frac{1}{c^2} \right ). \end{align} When powers of $$u$$ and $$v$$ that would otherwise describe exchanged $$\Delta = n \in \mathbb{N}$$ operators appear beside logs, we need to interpret this as a sign that the dimensions are really shifting.