  An algebraic variety is called $\mathbb{A}^{1}$-cylindrical if it contains an
$\mathbb{A}^{1}$-cylinder, i.e. a Zariski open subset of the form
$Z\times\mathbb{A}^{1}$ for some algebraic variety Z. We show that the generic
fiber of a family $f:X\rightarrow S$ of normal $\mathbb{A}^{1}$-cylindrical
varieties becomes $\mathbb{A}^{1}$-cylindrical after a finite extension of the
base. Our second result is a criterion for existence of an
$\mathbb{A}^{1}$-cylinder in X which we derive from a careful inspection of a
relative Minimal Model Program ran from a suitable smooth relative projective
model of X over S.
