Choose $o^t\in X^t$ so that $t\mapsto o^t$ is smooth and yet choose a famity of unit vectors $u^t\in T_{o^t}X^t$, so that $t\mapsto u^t$ is smooth.
Parametrize $X^t$ isothermaly by unit disc $f^t:D\to X^t$ in such a way that $f^t(0)=o^t$ and $(d_0f^t)(1)$ is proportional to $u^t$. Such a parametrization is unique. It follows that $t\mapsto f^t$ is continuous; otherwise different partial limits would give different isothermal coordinate with chousen origin and real direction.