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 $S$.