Consider a smooth hypersurface $T$ in a complex manifold $W$, and a submersion from $W$ onto a manifold $Y$ whose restriction to $T$ is also a submersion. In this paper we define the notion of a differential system specialisable along $T$ relative to the submersion ; this generalises the absolute case studied by Malgrange, Kashiwara, Mebkhout, and Sabbah. From this we deduce necessary and sufficient geometrical conditions for the existence of relative Bernstein polynomials for $mF^s$, $m$ being a section of a holonomic $\mathcal D$-module $\mathcal M$ and $F$ a holomorphic function on $W$. As an application we study the case when $\mathcal M$ is the local cohomology with support in a complete intersection.