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Let X/P¹ be a Mori fiber space with general fiber of Picard rank at least two. We prove that there is a proper closed subset S (X, invariant by the connected component of the identity Aut?(X) of the automorphism group of X, which is moreover the orbit of a section s and whose intersection with a fiber is an orbit of the subgroup of Aut?(X) acting trivially on P¹. Such result is a tool to describe equivariant birational maps from X/P¹ to other Mori fiber spaces and therefore finds its applications in the study of connected algebraic subgroups of Bir(X). This represents a first reduction step towards a possible classification of maximal connected algebraic subgroups of the Cremona group of rank 4. [ABSTRACT FROM AUTHOR]