On local time at time varying curve

Let $\left(X_{t} \right)_{t\geq 0}$ be a continuous semimartingale. Let $L^{z}_{t}\left(X\right)$ its family of local times. In \cite{YOR} Yor showed that the family $\left( L^{z}_{t}\left(X\right) \right)_{ z \in \mathbb{R}, t \geq 0}$ has a version that is continuous in $t$, c\`ad-l\`ag in $z$. In this paper we extend this result by showing that for a 'regular' family of curves parametrized by $z$, the corresponding family of local times has a 'regular' version. This result was used in \cite{BENT} to prove a change of variable formula for continuous semimartingales.