I am stuck with a simple example, but I guess the more general question would be whether blow ups commute with restrictions to subsets (points) of the blow-up locus.
Over $\mathbb{C}$, suppose that $\pi: X \rightarrow C$ is a family of singular (say, nodal) curves on $X$ parametrized by some smooth curve $C$. Suppose also that the singular locus of all fibers $X_c, c \in C$ is a curve $D$ on $X$. If I blow up $X$ along $D$, is every fiber of $\operatorname{Bl}_DX \rightarrow X \rightarrow C$ the blow up of $X_c$ along its singular point?
The example I looked at was the trivial family of nodal cubics $E \times \mathbb{A}^1 \rightarrow \mathbb{A}^1$, where $E$ is $y^2 = x^3+x^2$. Here - using coordinates $x, y, t$ - I wanted to blow up the total space along $\mathbb{A}^1 = V(x, y)$, which precisely contains the nodes of all fibers. I am stuck at describing the blow up and its fibers explicitly (rather, my computations don't end up with something sensible). Any help is greatly appreciated.
(This is more of a comment, but I don't have enough reputation yet.)
The basic problem is that blowing up is typically something you do to a subset of codimension two. (OK, there are subtleties involving blowing up non-Cartier divisors, but that's not what's going on here.)
But in your example, the fibres $X_c$ and the total space $X$ have singularities in codimension one. So to resolve the singularities, you should normalize first. The question is then whether the normalization of $X$ resolves each fibre. In your example, the answer is clearly yes. One could imagine more complicated examples, though, where the general fibre has several singular points which collide on the special fibre. In such a case I don't know what could happen.