  In this paper, we consider functionals related to mean curvature flow in an
ambient space which evolves by an extended Ricci flow from the perspective
introduced by Lott when studying a mean curvature flow in a Ricci flow
background. One of them is a weighted extended version of the
Gibbons-Hawking-York action on Riemannian metrics in compact manifolds with
boundary. We compute its variational properties from which naturally arise
boundary conditions to the analysis of its time-derivative under Perelman's
modified extended Ricci flow. For instance, the boundary integrand term
provides an extension of Hamilton's differential Harnack expression for mean
curvature flows in Euclidean space. We also derive the evolution equations for
both the second fundamental form and the mean curvature under mean curvature
flow in an extended Ricci flow background. In the special case of gradient
solitons to the extended Ricci flow, we discuss mean curvature solitons and
establish a Huisken's monotonicity-type formula. We show how to construct a
family of mean curvature solitons and establish a characterization of such a
family. Also, we show how for constructing examples of mean curvature solitons
in an extended Ricci flow background.
