  We present the first results on the complexity of the reconfiguration of
vertex separators under the three most popular rules: token addition/removal,
token jumping, and token sliding. We show that, aside from some trivially
negative instances, the first two rules are equivalent to each other and that,
even if only on a subclass of bipartite graphs, TJ is not equivalent to the
other two unless $\mathsf{NP} = \mathsf{PSPACE}$; we do this by showing a
relationship between separators and independent sets in this subclass of
bipartite graphs. In terms of polynomial time algorithms, we show that every
class with a polynomially bounded number of minimal vertex separators admits an
efficient algorithm under token jumping, then turn our attention to two classes
that do not meet this condition: $\{3P_1, diamond\}$-free and series-parallel
graphs. For the first, we describe a novel characterization, which we use to
show that reconfiguring vertex separators under token jumping is always
possible and that, under token sliding, it can be done in polynomial time; for
series-parallel graphs, we also prove that reconfiguration is always possible
under TJ and exhibit a polynomial time algorithm to construct the
reconfiguration sequence.
