  We recall the notion of (vertical) translating solitons in a product of a
semi-Riemannian manifold $(M,g)$ and the real line. Mainly, we restrict our
attention to those which are the graph of a smooth function. When dealing with
submersions, we show a criteria to lift (or project) translating solitons from
the base manifold to the total space (or viceversa). In particular, manifolds
foliated by codimension 1 orbits of a Lie group action give rise to such
solitons, up to solving a first-order ordinary differential equation. This
gives us explicit criteria under which the graph of a function is a soliton,
and we employ them to construct many examples of solitons, both new and old, in
a unified way.
