Let G be a connected algebraic group and X a variety endowed with a regular action of G and a Mori fibre space X/P^1 whose fibre is a Fano variety of Picard rank at least 2. In this talk I will explain why there is a proper horizontal subvariety of X which is invariant under the action of G, alongside with some applications of this result to the classification of connected algebraic subgroups of the Cremona group in dimension 4. This is a joint work with Jérémy Blanc.