In an earlier paper, the first two authors have shown that the convolution of a function $f$ continuous on the closure of a Cartan domain and a $K$-invariant finite measure $\mu$ on that domain is again continuous on the closure, and, moreover, its restriction to any boundary face $F$ depends only on the restriction of $f$ to $F$ and is equal to the convolution, in $F$, of the latter restriction with some measure ${\mu }_F$ on $F$ uniquely determined by $\mu$. In this article, we give an explicit formula for ${\mu }_F$ in terms of $F$,...