Let $k$ be a field of characteristic zero, $R=k[[ t ]] $ the ring of formal power series and $K=k(( t)) $ its fraction field. Let $X$ be a finite type $R$-scheme with smooth generic fiber. Let $\mathcal X$ be the $t$-adic completion of $X$ and $\mathcal X_{\eta }$ the generic fiber of $\mathcal X$. Let $Z\subset X_\sigma $ be a locally closed subset of the special fiber of $X$. In this article, we establish a relation between the rigid motive of $]Z[$ (the tube of $Z$ in $\mathcal X_\eta $) and the restriction to $Z$ of the nearby motivic sheaf associated with the $R$-scheme $X$. Our main result, Theorem ??, can be interpreted as a motivic analog of a theorem of Berkovich. As an application, given a rational point $x\in X_{\sigma }$, we obtain an equality, in a suitable Grothendieck ring of motives, between the motivic Milnor fiber of Denef-Loeser at $x$ and the of the rigid motive of the analytic Milnor fiber of Nicaise-Sebag at $x$.