  We analyse the $r \to \infty$ asymptotic decay of the total correlation
function, $h(r)$, for a fluid composed of particles interacting via a (point)
Yukawa pair potential. Such a potential provides a simple model for dusty
plasmas. The asymptotic decay is determined by the poles of the liquid
structure factor in the complex plane. We use the hypernetted-chain closure to
the Ornstein-Zernike equation to determine the line in the phase diagram,
well-removed from the freezing transition line, where crossover occurs in the
ultimate decay of $h(r)$, from monotonic to damped oscillatory. We show: i)
crossover takes place via the same mechanism (coalescence of imaginary poles)
as in the classical one-component plasma and in other models of Coulomb fluids
and ii) leading-order pole contributions provide an accurate description of
$h(r)$ at intermediate distances $r$ as well as at long range.
