An elimination tree of a connected graph $G$ is a rooted tree on the vertices of $G$ obtained by choosing a root $v$ and recursing on the connected components of $G-v$ to obtain the subtrees of $v$. The graph associahedron of $G$ is a polytope whose vertices correspond to elimination trees of $G$ and whose edges correspond to tree rotations, a natural operation between elimination trees. These objects generalize associahedra, which correspond to the case where $G$ is a path. Ito et al. [ICALP 2023] recently proved that the problem of computing distances on graph associahedra is NP-hard. In this paper we prove that the problem, for a general graph $G$, is fixed-parameter tractable parameterized by the distance $k$. Prior to our work, only the case where $G$ is a path was known to be fixed-parameter tractable. To prove our result, we use a novel approach based on a marking scheme that restricts the search to a set of vertices whose size is bounded by a (large) function of $k$. On the negative side, we show that it is unlikely that FPT algorithms exist on a natural generalization of graph associahedra, namely hypergraphic polytopes, by proving that computing distances on them is W[2]-hard parameterized by the distance. We also prove that, on hypergraphic polytopes, the distance cannot be approximated in polynomial time within a factor $c \cdot \log(|V|+|\mathcal{E}|)$ for some constant $c > 0$ unless P = NP, where $H=(V, \mathcal{E})$ is the input hypergraph. This result strengthens the hardness result of Cardinal and Steiner [Combin. Theory 2025], who proved that the problem cannot be approximated within a factor $(1 + \varepsilon)$ for some absolute constant $\varepsilon > 0$ unless P = NP. Finally, we rule out the existence of polynomial kernels parameterized by the number of vertices of the input hypergraph, a parameter for which the problem is easily seen to be FPT.