  A surgery of a real symplectic manifold $X_{\mathbb R}$ along a real
Lagrangian sphere $S$ is a modification of the symplectic and real structure on
$X_{\mathbb R}$ in a neigborhood of $S$. Genus 0 Welschinger invariants of two
real symplectic $4$-manifolds differing by such a surgery have been related in
a previous work in collaboration with N. Puignau. In the present paper, we
explore some particular situations where these general formulas greatly
simplify. As an application, we complete the computation of genus 0 Welschinger
invariants of all del~Pezzo surfaces, and of all $\mathbb R$-minimal real conic
bundles. As a by-product, we establish the existence of some new relative
Welschinger invariants. We also generalize our results to the enumeration of
curves of higher genus, and give relations between hypothetical invariants
defined in the same vein as a previous work by Shustin.
