Let $X$ and $Y$ be smooth projective complex varieties. Suppose we have a Fourier-Mukai equivalence $$ \Phi_\mathcal P :Perf X \to Perf Y $$ with kernel $\mathcal P$. Moreover, suppose $\mathcal P$ locally corresponds to a graph of a birational map $\phi:X \supset U \cong V \subset Y$. Now, if there is a (necessarily unique) extension of $\phi$ to a larger open subset $U\subsetneq U'\cong V' \supsetneq V$, is it true that $\mathcal P$ still locally corresponds to the graph of the extended birational map? I am particaularly interested in the case when $\phi$ extends to an isomorphism $X\cong Y$ and wondering if there is a well-known counter example.