Cox rings of blow-ups of multiprojective spaces

Let $X_r^{1,n}$ be the blow-up of ${\mathbb{P}}^1\times {\mathbb{P}}^n$ in $r$ general points. We describe the Mori cone of $X_r^{1,n}$ for $r\le n+2$ and for $r=n+3$ when $n\le 4$. Furthermore, we prove that $X_{n+1}^{1,n}$ is log Fano and give an explicit presentation for its Cox ring.