  This paper is devoted to the analysis of Steklov eigenvalues and Steklov
eigenfunctions on a class of warped product Riemannian manifolds $(M,g)$ whose
boundary $\partial M$ consists in two distinct connected components $\Gamma_0$
and $\Gamma_1$. First, we show that the Steklov eigenvalues can be divided into
two families $(\lambda_m^\pm)_{m \geq 0}$ which satisfy accurate asymptotics as
$m \to \infty$. Second, we consider the associated Steklov eigenfunctions which
are the harmonic extensions of the boundary Dirichlet to Neumann
eigenfunctions. In the case of symmetric warped product, we prove that the
Steklov eigenfunctions are exponentially localized on the whole boundary
$\partial M$ as $m \to \infty$. Whenever we add an asymmetric perturbation to a
symmetric warped product, we observe a flea on the elephant effect. Roughly
speaking, we prove that "half" the Steklov eigenfunctions are exponentially
localized on one connected component of the boundary, say $\Gamma_0$, and the
other half on the other connected component $\Gamma_1$ as $m \to \infty$.
