  A matching $M$ in a graph $G$ is connected if all the edges of $M$ are in the
same component of $G$. Following \L uczak,there have been many results using
the existence of large connected matchings in cluster graphs with respect to
regular partitions of large graphs to show the existence of long paths and
other structures in these graphs. We prove exact
  Ramsey-type bounds on the sizes of monochromatic connected matchings in
$2$-edge-colored multipartite graphs. In addition, we prove a stability theorem
for such matchings.
