The answer is **yes** when $\dim D \leq 2$ and the variety upstairs (i.e. $C$ in your notation) is *normal*: 
see

Bas Edixhoven, Robin de Jong, Jan Schepers, [*Covers of surfaces with fixed branch locus*][1], Lemma 2.1

for a proof in dimension $2$ (which also works in dimension $1$).

It seems plausible that this proof can be extended in any dimension $\geq 3$, although I did not check it carefully.



  


 


  [1]: http://arxiv.org/pdf/0807.0184v2.pdf