Pattern Equivariant Mass Transport in Aperiodic Tilings and Cohomology

Suppose that we have a repetitive and aperiodic tiling $\bf T$ of $\mathbb{R}^n$, and two mass distributions $f_1$ and $f_2$ on $\mathbb{R}^n$, each pattern equivariant with respect to $\bf T$. Under what circumstances is it possible to do a bounded transport from $f_1$ to $f_2$? When is it possible to do this transport in a strongly or weakly pattern-equivariant way? We reduce these questions to properties of the \v Cech cohomology of the hull of $\bf T$, properties that in most common examples are already well-understood.