Composition operators between model and Hardy spaces

Let $n\ge 1$ and $\phi :{\mathbb{D}}^n\to \mathbb{D}$ be a holomorphic function, where $\mathbb{D}$ denotes the open unit disk of $\mathbb{C}$. Let $\mathrm{\Theta }:\mathbb{D}\to \mathbb{D}$ be an inner function and $K_{\mathrm{\Theta }}^p$, $p>0$, denote the corresponding model space. We obtain characterizations of the compact composition operators $C_{\phi }:K_{\mathrm{\Theta }}^p\to H^p({\mathbb{D}}^n)$, $1<p<\mathrm{\infty }$, where $H^p({\mathbb{D}}^n)$ denotes the Hardy space.