  This article presents new parabolic and elliptic type gradient estimates for
positive smooth solutions to a nonlinear parabolic equation involving the
Witten Laplacian in the context of smooth metric measure spaces. The metric and
potential here are time dependent and evolve under a super Perelman-Ricci flow.
The estimates are derived under natural lower bounds on the associated
generalised Bakry-\'Emery Ricci curvature tensors and are utilised in
establishing fairly general local and global bounds, Harnack-type inequalities
and Liouville-type global constancy theorems to mention a few. Other
implications and consequences of the results are also discussed.
