  This paper investigates the expressiveness of a fragment of first-order
sentences in Gaifman normal form, namely the positive Boolean combinations of
basic local sentences. We show that they match exactly the first-order
sentences preserved under local elementary embeddings, thus providing a new
general preservation theorem and extending the L\'os-Tarski Theorem.
  This full preservation result fails as usual in the finite, and we show
furthermore that the naturally related decision problems are undecidable. In
the more restricted case of preservation under extensions, it nevertheless
yields new well-behaved classes of finite structures: we show that preservation
under extensions holds if and only if it holds locally.
