Let $\mathcal{M},\mathcal{F}$ be the classifiying spaces (i.e. complex manifolds) of two (possibly) different moduli problem. To give a map $$f:\mathcal{M}\rightarrow \mathcal{F}$$ means that for each object classified by $\mathcal{M}$, we must associate an object classified by $\mathcal{F}$. Now if I want to show that this map is holomorphic, then the way I've seen this often done is that you need to use some explicit local parametrization of both $\mathcal{M}$ and $\mathcal{F}$.
However, I wonder if this can not be done in a deformation theoretic way. Intuitively, infinitesimal deformations of an object $m\in\mathcal{M}$ corresponds to the tangent space of our manifold, and if we assume that our moduli space is smooth, it would thus also correspond to a "small" neighborhood of $m$. Since we only need to show that in sufficienttly small neighborhoods of point, the map is a power series, I feel like that the stalk $\mathcal{O}_{\mathcal{M},m}$ should "see" the holomorphicity.
As a concrete example, consider the moduli space of surfaces $\mathcal{M}_{g,n}$ and the flag variety $\mathcal{Fl}$. To each $[X,x_1,\ldots,x_n]\in \mathcal{M}_{g,n}$ we can associate the flag $F^p(H^k(X,\mathbb{C}))$. It is known by the work of Griffiths that this map is holomorphic. Can this be proven by studying how the period map behaves at the Teichmüller space $\mathcal{T}_g$?
