Let $X$ and $Y$ be smooth projective geometrically connected curves over $k$ of genus $g$ at least two.

If $k$ is an algebraically closed field of characteristic zero, there exists a connected variety $T$ over $k$, points $x,y \in T(k)$ and a family of curves $\mathcal C\to T$ such that $\mathcal C_{t_0} = X$ and $\mathcal C_{t_1} = Y$. In other words, there is a deformation space over $k$ which deforms $X$ into $Y$.

My question is whether such a space exists if $k$ it not algebraically closed.

Suppose that $k = \mathbb Q$. Does there exist a deformation from $X$ to $Y$ over $k$?

More precisely, does there exist a connected variety $T$ over $\mathbb Q$, points $x,y\in T(k)$ and a family of curves $\mathcal C\to T$ such that $\mathcal C_{t_0} = X$ and $\mathcal C_{t_1} = Y$?

When $X$ and $Y$ have level $n$ structure (for some $n>2$), then you can take $T$ to be the moduli space of curves with level n structure over $\mathbb Q$. Also, without any assumption of level structure, it is clear that there exists such a deformation space over a finite extension of $\mathbb Q$. But does it exist over $\mathbb Q$?

My guess would be that such a space doesn't exist in general.