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, 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.