Let $p:Z\to X$ be a continuous surjective map between compact Hausdorff spaces. Does there exist a family $m=(m_x)_{x\in X}$ of Radon probability measures on $Z$, such that

the support of $m_x$ is contained in the fibre $p^{-1}(\{ x\})$,

for every $f\in C(Z)$ we have $f^m\in C(X)$, where $f^m(x)=\int_Zf(z)\ dm_x(z)$.

If not, under which additional restrictions does it exist? I am particularly interested in the case, when $Z$ is not first countable.