  We develop a gradient-flow theory for time-dependent functionals defined in
abstract metric spaces. Global well-posedness and asymptotic behavior of
solutions are provided. Conditions on functionals and metric spaces allow to
consider the Wasserstein space $\mathscr{P}_{2}(\mathbb{R}^{d})$ and apply the
results for a large class of PDEs with time- dependent coefficients like
confinement and interaction potentials and diffusion. Our results can be seen
as an extension of those in Ambrosio-Gigli-Savar\'e (2005)[2] to the case of
time-dependent functionals. For that matter, we need to consider some residual
terms, time-versions of concepts like $\lambda$-convexity,
time-differentiability of minimizers for Moreau-Yosida approximations, and a
priori estimates with explicit time-dependence for De Giorgi interpolation.
Here, functionals can be unbounded from below and satisfy a type of
$\lambda$-convexity that changes as the time evolves.
