We give a description of the algebraic families of birational transformations of an algebraic variety X. As an application, we show that the morphisms to Bir(X) given by algebraic families satisfy a Chevalley type result and a certain fibre-dimension formula. Moreover, we show that the algebraic subgroups of Bir(X) are exactly the closed finite-dimensional subgroups with finitely many components. We also study algebraic families of birational transformations preserving a fibration. This builds on previous work of Blanc-Furter 2, Hanamura 9, and Ramanujam 20.
Regeta et al. (Wed,) studied this question.