I could not decide if I should post this question in MO or Mathstackexchange, so feel free to downvote it if you think it does not belong here. I will delete my post and post it in MathSE in that case.

What I am wondering about is the following: Given a projective map $\pi:X\to B$, where $B$ is integral and $\mathscr{F}$ a coherent sheaf on $X$, which is flat over $Y$. Grauert's theorem tells us that there is an identification $$R^i\pi_*(\mathscr{F})\otimes k(b)\cong H^i(X_b,\mathscr{F}_b),$$ provided that the dimension of $H^i(X_b,\mathscr{F}_b)$ is the same for all $b\in B$. I want to know what happens if we relax the last condition.

To this end, let us consider the following example: Let $C$ be a genus $g$ curve and $\mathscr{L}$ a Poincare line bundle on $C\times Pic^{2g-2}(C)$ and consider ${\pi_2}_*\mathscr{L}$ on $Pic^{2g-2}(C)$. Now we know that over the open set $U$ of nonspecial bundles in $Pic^{2g-2}(C)$ ,we have (by Grauert's theorem) the identification that $${\pi_2}_*\mathscr{L}\otimes k(L)\cong H^0(C,L)$$ for every $L\in U$. What happens in this case over special bundles? For instance, what is $${\pi_2}_*\mathscr{L}\otimes k(K_C)$$ concretely and how far is it from $H^0(C,K_C)$? ($K_C$ being of course the canonical bundle of $C$)