We present a deformation of surfaces from a product space $M_1\times\mathbb{R}$ into another product space $M_2\times\mathbb{R}$ such that the relation of the principal curvatures of the deformed surfaces can be controlled in terms of the curvatures of $M_1$ and $M_2$. Thus, starting from a known example, we obtain subsolutions for the existence or barriers for the non existence of surfaces with fixed mean curvature, extrinsic curvature or Gaussian curvature in $M\times\mathbb{R}$.