  For dry foams, the transport of gas from small high-pressure bubbles to large
low-pressure bubbles is dominated by diffusion across the thin soap films
separating neighboring bubbles. For wetter foams, the film areas become smaller
as the Plateau borders and vertices inflate with liquid. So-called
"border-blocking" models can explain some features of wet-foam coarsening based
on the presumption that the inflated borders totally block the gas flux,
however, this approximation dramatically fails in the wet/unjamming limit where
the bubbles become close-packed spheres and coarsening proceeds even though
there are no films. Here, we account for the ever-present border-crossing flux
by a new length scale defined by the average gradient of gas concentration
inside the borders. We compute that it is proportional to the geometric average
of film and border thicknesses, and we verify this scaling by numerical
solution of the diffusion equation. We similarly consider transport across
inflated vertices and surface Plateau borders in quasi-2d foams. And we show
how the $dA/dt=K_0(n-6)$ von~Neumann law is modified by the appearance of terms
that depend on bubble size and shape as well as the concentration gradient
length scales. Finally, we use the modified von~Neumann law to compute the
growth rate of the average bubble area, which is not constant.
