  In this paper, we mainly investigate continuity, monotonicity and
differentiability for the first eigenvalue of the $p$-Laplace operator along
the Ricci flow on closed manifolds. We show that the first $p$-eigenvalue is
strictly increasing and differentiable almost everywhere along the Ricci flow
under some curvature assumptions. In particular, for an orientable closed
surface, we construct various monotonic quantities and prove that the first
$p$-eigenvalue is differentiable almost everywhere along the Ricci flow without
any curvature assumption, and therefore derive a $p$-eigenvalue comparison-type
theorem when its Euler characteristic is negative.
