This question is somehow a more generalized re-edit of a former question of mine:
I guess and hope that this will make the question clearer.
Suppose I have a flat family $X \rightarrow F$ of geometric objects (vector bundles, curves, etc) and a GIT quotients $M$ that gives a coarse moduli space for the same moduli problem. Supose further that the image of $F$ (that I will call $F$ as well, abusing notation, and I suppose smooth) under the classifying map intersect the singular, strictly semi-stable locus of $M$.
Suppose now I blow up $F$ along $F\cap M$ - we can even assume $F\cap M$ is a point $p$, as long it is cod 2. Let me denote by $FF$ del blown-up variety and $E$ the exceptional divisor..
In my particular case, over the exceptional divisor there's a natural family of objects $Z\rightarrow E$ and $E$ that is contracted to $F\cap M$ by the classifying map (I am quite sure that this is not always the case). So one can see the blow down of $E$ as a modular, universal, natural map. In particular one of the objects over $E$ is isomorphic to $X_p$, the object of $F$ over the singular point.
The question is: can I expect the existence of a flat family $Y$ over $FF$ s.t. the restriction to $E$ is iso to $Z$ and the restriction to $FF/E$ is iso to $X$?