  In this article, we study the powers of the generalized binomial edge ideal
$\mathcal{J}_{K_m,P_n}$ of a path graph $P_n$. We explicitly compute their
regularities and determine the limit of their depths. We also show that these
ordinary powers coincide with their symbolic powers. Additionally, we study the
Rees algebra and the special fiber ring of $\mathcal{J}_{K_m,P_n}$ via Sagbi
basis theory. In particular, we obtain exact formulas for the regularity of
these blowup algebras.
