  Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set
$X$ and diameter $D$. Let $A$ denote the adjacency matrix of $\Gamma$. Fix a
base vertex $x\in X$ and for $0 \leq i \leq D$ let $E^*_i=E^*_i(x)$ denote the
projection matrix to the $i$th subconstituent space of $\Gamma$ with respect to
$x$. The Terwilliger algebra $T(x)$ of $\Gamma$ with respect to $x$ is the
semisimple subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by $A, E^*_0,
E^*_1, \ldots, E^*_D$. Remark that the isomorphism class of $T(x)$ depends on
the choice of the base vertex $x$. We say $\Gamma$ is pseudo-vertex-transitive
whenever for any vertices $x,y \in X$, the Terwilliger algebras $T(x)$ and
$T(y)$ are isomorphic. In this paper we discuss pseudo-vertex transitivity for
distance-regular graphs with diameter $D\in \{2,3,4\}$. In the case of diameter
$2$, a strongly regular graph $\Gamma$ is thin, and $\Gamma$ is
pseudo-vertex-transitive if and only if every local graph of $\Gamma$ has the
same spectrum. In the case of diameter $3$, Taylor graphs are thin and
pseudo-vertex-transitive. In the case of diameter $4$, antipodal tight graphs
are thin and pseudo-vertex-transitive.
