  Let $k$ be a field and $g \colon Y \longrightarrow X$, $f \colon X
\longrightarrow \mathbb{A}_k^1$ be morphisms of quasi-projective $k$-varieties.
In his thesis, Ayoub developed the theory of motivic nearby cycles functors
$\Psi_{f},\Psi_{f \circ g}$. They are equipped with several base change
morphisms such as $\mu_g \colon g_{\sigma!}\Psi_{f \circ g} \longrightarrow
\Psi_f g_{\eta !}$, $\nu_g\colon \Psi_{f \circ g}g_{\eta}^! \longrightarrow
g_{\sigma}^!\Psi_f$, which are isomorphisms if $g$ is projective and smooth,
respectively. In this article, we remove the quasi-projectiveness of varieties
in these base change morphisms by following a classical formulation of Deligne.
As an application, we prove that the cone of $\mu_g$ is a triangulated functor.
It turns out that nearby cycles at infinity constructed by Raibaut is
incarnation of this cone functor.
