I stumbled upon the following question for families of vector bundles (over a curve) but I guess it could be interesting to answer in general.
Suppose I have $B$ the blow-up of a smooth projective variety $M$ along a subvariety $N$. Let $E$ be the exceptional divisor over $N$. Suppose that $E$ has a section (or better that the normal bundle of $N$ has a nonvanishing section), then we can identify $N$ with the image of such section.
Then, suppose I have two flat families of objects: one $\Xi$ over $M$ and the other $\Psi$ over $E$, and suppose that they agree on $N$. Under what condition there exists a universal flat family $\Phi$ over $B$, whose restriction to $M$ (resp. to $E$) is equal to $\Xi$ (resp. to $\Psi$)?