  We show that the generic fiber of a family of smooth $\mathbb{A}^{1}$-ruled
affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a
finite extension of the base. In the particular case where the general fibers
of the family are irrational surfaces, we establish that up to shrinking the
base, such a family actually factors through an $\mathbb{A}^{1}$-fibration over
a certain scheme, induced by the MRC-fibration of a relative smooth projective
model of the family. For affine threefolds fibered by irrational
$\mathbb{A}^{1}$-ruled surfaces, this induced $\mathbb{A}^{1}$-fibration can
also be obtained from a relative Minimal Model Program applied to a relative
smooth projective model of the family.
