Isomorphism of locally compact Polish metric structures

Gao and Kechris asked if the isometry relation on the space of locally compact Polish metric spaces is (Borel) reducible to the isomorphism relation of countable graphs. It turns out that this question is easier to deal with if it is translated to the language of metric structures and continuous logic. I prove that isomorphism on Borel classes of locally compact Polish metric structures is reducible to graph isomorphism, which immediately gives a positive answer to Gao and Kechris' question. I also analyse Borel isomorphism relations on such classes in terms of their reducibility to equality on hereditarily countable sets of appropriately bounded rank.