Jenkins-Serrin problem for translating horizontal graphs in $M\times \mathbb{R}$

We prove the existence of horizontal Jenkins-Serrin graphs that are translating solitons of the mean curvature flow in Riemannian product manifolds $M\times \mathbb{R}$. Moreover, we give examples of these graphs in the cases of ${\mathbb{R}}^3$ and ${\mathbb{H}}^2\times \mathbb{R}$.