Let $A = \overline{\bigcup A_n}$, $B = \overline{\bigcup B_n}$ be AF algebras with the same Bratelli diagram. Then there is an isomorphism $\phi : A \rightarrow B$.
Let $x \in \bigcup A_n$ (say, $x \in A_k$ for some $k$). Is it always the case that $\phi(x) \in \bigcup B_n$ ?
If you look at the proof of Proposition III.2.7 in Davidson's C$^*$-algebras by Example, the isomorphism is constructed first between $\bigcup A_n$ and $\bigcup B_n$, and only then extended to the closures (using that it is isometric).