OK, I hope you don't mind if I edited.

Be careful, if $X$ is an hypersurface $\pi$ is far from being birational!

In fact such an example is not possible for curves, where rationality correspond to genus 0, and the genus is constant over flat families.

Thanks for everybody's contribution. I meant exactly what Jason and Sandor say. Could you shortly explain me why? I also expected this to exist, but could not show this.

Let us supppose it does, just to start with an easier hypothesis.

It would be useful to know some more on $Y$. Is it a smooth scheme?

I think so. Take a general genus 4 curve, embedded canonically in $\mathbb{P}^3$ as a $(2,3)$ complete intersection. Take $\mathcal{L}=\mathcal{O}$, $k=1$ and $n=2$. Then $h^0(\mathcal{O}(3))=20$, the multiplication map seems to have Kernel of dimension 4.

Yeah, both your answer are very nice. It is clear that GRR is the way to get it right. I seem to understand anyway that it would be far too optimistic to expect a quick formula relating the two sheaves.

Good point. Yes $\pi_{!}\mathcal{S}$ would be ok, I edit the question. I would be curious to see even in the derived category.

What are the assumptions on $X$? Is it a scheme or what?