  It is a theorem of Kim-Tamagawa that the $\mathbb Q_\ell$-pro-unipotent
Kummer map associated to a smooth projective curve $Y$ over a finite extension
of $\mathbb Q_p$ is locally constant when $\ell\neq p$. The present paper
establishes two generalisations of this result. Firstly, we extend the
Kim-Tamagawa Theorem to the case that $Y$ is a smooth variety of any dimension.
Secondly, we formulate and prove the analogue of the Kim-Tamagawa Theorem in
the case $\ell = p$, again in arbitrary dimension. In the course of proving the
latter, we give a proof of an \'etale-de Rham comparison theorem for
pro-unipotent fundamental groupoids using methods of Scholze and
Diao-Lan-Liu-Zhu. This extends the comparison theorem proved by Vologodsky for
certain truncations of the fundamental groupoids.
