  In this paper we show that certain almost distance-regular graphs, the
so-called $h$-punctually walk-regular graphs, can be characterized through the
cospectrality of their perturbed graphs. A graph $G$ with diameter $D$ is
called $h$-punctually walk-regular, for a given $h\le D$, if the number of
paths of length $\ell$ between a pair of vertices $u,v$ at distance $h$ depends
only on $\ell$. The graph perturbations considered here are deleting a vertex,
adding a loop, adding a pendant edge, adding/removing an edge, amalgamating
vertices, and adding a bridging vertex. We show that for walk-regular graphs
some of these operations are equivalent, in the sense that one perturbation
produces cospectral graphs if and only if the others do. Our study is based on
the theory of graph perturbations developed by Cvetkovi\'c, Godsil, McKay,
Rowlinson, Schwenk, and others. As a consequence, some new characterizations of
distance-regular graphs are obtained.
