We show that if $R$ is a local domain which is dominated by a valuation $v$, then there does not always exist a regular local ring $R^{\prime }$ which birationally dominates $R$ and is dominated by $v$ and an extension of $v$ to the Henselization $(R^{\prime }{)}^h$ of $R^{\prime }$ such that the associated graded rings of $R^{\prime }$ and $(R^{\prime }{)}^h$ along the valuations are equal. We also show that there does not always exist $R^{\prime }$, a prime ideal p of the completion of $R^{\prime }$ such that $p^{\prime }\cap R^{\prime }=(0)$ and an extension of $v$ to $R^{\prime }$ such that the associated graded rings of $R^{\prime }$ and $R^{\prime }/p$ along the valuation are equal.