Isoperimetric minimizing movements and AC curves in $\text{PL}_q^p(\mathbb{R}^n)$

We define a complete metric structure $\mathfrak{d}_q^p$ on the family $\text{PL}_q^p(\mathbb{R}^n)$ of probability measures with densities in $L^p(\mathbb{R}^n)$ and finite $q$-moments. We establish the existence of generalized minimizing movements for the isoperimetric ratio and characterize $\mathfrak{d}_q^p$-absolutely continuous curves through weak solutions of the continuity equation with velocity fields satisfying a Sobolev-type condition. We also characterize absolutely continuous curves in the $\infty$-Wasserstein space and prove a Benamou--Brenier formula for $W_\infty$.