  Let $\mathbb{E}$ be a connected and orientable Riemannian 3-manifold with a
non-singular Killing vector field $\xi\in\mathfrak{X}(\mathbb{E})$ whose
associated one-parameter group of the isometries of $\mathbb{E}$ acts freely
and properly on $\mathbb{E}$. Then, there is a Killing submersion from
$\mathbb{E}$ onto a connected and orientable surface $M$ whose fibers are the
integral curves of $\xi$. In this setting we solve the Dirichlet problem for
minimal Killing graphs over certain unbounded domains of $M$, taking piecewise
continuous boundary values. We obtain general Collin-Krust type estimates. In
the particular case of the Heisenberg group, we prove a uniqueness result for
minimal Killing graphs with bounded boundary values over a strip. We also prove
that isolated singularities of Killing graphs with prescribed mean curvature
are removable.
