  The goal in {\em reconfiguration problems} is to compute a {\em gradual
transformation} between two feasible solutions of a problem such that all
intermediate solutions are also feasible. In the {\em Matching Reconfiguration
Problem} (MRP), proposed in a pioneering work by Ito et al.\ from 2008, we are
given a graph $G$ and two matchings $M$ and $M'$, and we are asked whether
there is a sequence of matchings in $G$ starting with $M$ and ending at $M'$,
each resulting from the previous one by either adding or deleting a single edge
in $G$, without ever going through a matching of size $< \min\{|M|,|M'|\}-1$.
Ito et al.\ gave a polynomial time algorithm for the problem.
  In this paper we introduce a natural generalization of the MRP that depends
on an integer parameter $\Delta \ge 1$: here we are allowed to make $\Delta$
changes to the current solution rather than 1 at each step of the
{transformation procedure}. There is always a valid sequence of matchings
transforming $M$ to $M'$ if $\Delta$ is sufficiently large, and naturally we
would like to minimize $\Delta$. We first devise an optimal transformation
procedure for unweighted matching with $\Delta = 3$, and then extend it to
weighted matchings to achieve asymptotically optimal guarantees. The running
time of these procedures is linear.
  We further demonstrate the applicability of this generalized problem to
dynamic graph matchings. In this area, the number of changes to the maintained
matching per update step (the \emph{recourse bound}) is an important quality
measure. Nevertheless, the \emph{worst-case} recourse bounds of almost all
known dynamic matching algorithms are prohibitively large, much larger than the
corresponding update times. We fill in this gap via a surprisingly simple
black-box reduction: Any dynamic algorithm for maintaining [...]
