On non-rigid del Pezzo fibrations of low degree

We consider $\mathbb{P}(1,1,1,2)$ bundles over ${\mathbb{P}}^1$ and construct hypersurfaces of these bundles which form a degree 2 del Pezzo fibration over ${\mathbb{P}}^1$ as a Mori fibre space. We classify all such hypersurfaces whose type $\text{III}$ or $\text{IV}$ Sarkisov links pass to a different Mori fibre space. A similar result for cubic surface fibrations over ${\mathbb{P}}^2$ is also presented.