  In this paper we obtain a Hadamard type formula for simple eigenvalues and an
analog to the Rayleigh-Faber-Krahn inequality for a class of nonlocal
eigenvalue problems. Such class of equations include among others, the
classical nonlocal problems with Dirichlet and Neumann conditions. The Hadamard
formula is computed allowing domain perturbations given by embeddings of
$n$-dimensional Riemannian manifolds (possibly with boundary) of finite volume
while the Rayleigh-Faber-Krahn inequality is shown by rearrangement techniques.
