  We study the asymptotic Dirichlet problem for Killing graphs with prescribed
mean curvature $H$ in warped product manifolds $M\times_\varrho \mathbb{R}$. In
the first part of the paper, we prove the existence of Killing graphs with
prescribed boundary on geodesic balls under suitable assumptions on $H$ and the
mean curvature of the Killing cylinders over geodesic spheres. In the process
we obtain a uniform interior gradient estimate improving previous results by
Dajczer and de Lira. In the second part we solve the asymptotic Dirichlet
problem in a large class of manifolds whose sectional curvatures are allowed to
go to $0$ or to $-\infty$ provided that $H$satisfies certain bounds with
respect to the sectional curvatures of $M$ and the norm of the Killing vector
field. Finally we obtain non-existence results if the prescribed mean curvature
function $H$ grows too fast.
