<p>We study the variation of the Enriquez connection for higher genus polylogarithms under degenerations of Riemann surfaces with marked points, and show that this connection becomes the connection constructed by the author for degenerating families of pointed Riemann surfaces. Therefore, we have an important application that the higher genus polylogarithms derived from the Enriquez connection can be described explicitly as power series in deformation parameters and their logarithms associated with the families whose coefficients are expressed by multiple zeta values.</p>