  We analyze certain compositions of rational inner functions in the unit
polydisk $\mathbb{D}^{d}$ with polydegree $(n,1)$, $n\in \mathbb{N}^{d-1}$, and
isolated singularities in $\mathbb{T}^d$. Provided an irreducibility condition
is met, such a composition is shown to be a rational inner function with
singularities in precisely the same location as those of the initial function,
and with quantitatively controlled properties. As an application, we answer a
$d$-dimensional version of a question posed in \cite{BPS22} in the affirmative.
