  We study the connected algebraic groups acting on Mori fibrations $X \to Y$
with $X$ a rational threefold and $\mathrm{dim}(Y) \geq 1$. More precisely, for
these fibre spaces we consider the neutral component of their automorphism
groups and study their equivariant birational geometry. This is done using,
inter alia, minimal model program and Sarkisov program and allows us to
determine the maximal connected algebraic subgroups of
$\mathrm{Bir}(\mathbb{P}^3)$, recovering most of the classification results of
Hiroshi Umemura in the complex case.
