Cuspidal $\ell$-modular representations of $p$-adic classical groups

For any algebraically closed field $R$ of characteristic $\ell$, we construct all irreducible cuspidal $R$-representations of a $p$-adic classical group G, with $p$ different from $\ell$, extending the second authors construction when $R=\mathbb{C}$. We show a relationship between this construction and the construction of level zero irreducible cuspidal $R$-representations in an associated product of classical and general linear groups, allowing for an initial refinement of the exhaustive list, which is also new for $\mathbb{C}$-representations.