  Given a graph and two vertex sets satisfying a certain feasibility condition,
a reconfiguration problem asks whether we can reach one vertex set from the
other by repeating prescribed modification steps while maintaining feasibility.
In this setting, Mouawad et al. [IPEC 2014] presented an algorithmic
meta-theorem for reconfiguration problems that says if the feasibility can be
expressed in monadic second-order logic (MSO), then the problem is
fixed-parameter tractable parameterized by $\textrm{treewidth} + \ell$, where
$\ell$ is the number of steps allowed to reach the target set. On the other
hand, it is shown by Wrochna [J. Comput. Syst. Sci. 2018] that if $\ell$ is not
part of the parameter, then the problem is PSPACE-complete even on graphs of
bounded bandwidth.
  In this paper, we present the first algorithmic meta-theorems for the case
where $\ell$ is not part of the parameter, using some structural graph
parameters incomparable with bandwidth. We show that if the feasibility is
defined in MSO, then the reconfiguration problem under the so-called token
jumping rule is fixed-parameter tractable parameterized by neighborhood
diversity. We also show that the problem is fixed-parameter tractable
parameterized by $\textrm{treedepth} + k$, where $k$ is the size of sets being
transformed. We finally complement the positive result for treedepth by showing
that the problem is PSPACE-complete on forests of depth $3$.
