Let $\pi \colon \mathfrak{X} \rightarrow B$ be a deformation of complex compact manifolds and $E$ be a holomorphic vector bundle on $\mathfrak{X}$ (or a coherent sheaf on $\mathfrak{X}$ that is flat over $B$).
The function $b \rightarrow \mathrm{h}^0(X\_b, E\_{|X\_b})$ is known to be upper semi-continuous, but I wonder if it is constructible. 


If the deformation is algebraic, that is if there exists a relatively ample line bundle on $\mathfrak{X}$, then there exists a complex $\mathcal{L}^{\bullet}$ of locally free sheaves on ${B}$ such that for any $b$ in $B$ and any nonnegative integer $i$, 
$$
\mathrm{H}^i(\mathcal{L}^{\bullet}\_{|b}) \simeq \mathrm{H}^i(X\_b, E\_{| X\_b}).
$$
This is proved for instance in [Voisin, Complex algebraic geometry, Vol. 1, section 9.3.1], and implies constructibility. 


Is the same result known in the complex analytic case? 



Cohomology of vector bundles in families
The constructibility result you ask for is Satz 7.7(1) in [Ein Kriterium für die Offenheit der Versalität (Flenner, 1981)](http://dx.doi.org/10.1007/BF01174768).


The appropriate generalization of the result in Voisin's book that you mention is the topic of [Eine Bemerkung über relative Ext-Garben (Flenner, 1981)](http://dx.doi.org/10.1007/BF01450533).


