Let $W = C \times_Y X$. I imagine $f^*C$ is suitably interpreted as the chow class on $W$ given by $C \cdot X$. Write $i : C \subseteq Y$, $f': W \to C$, $i' : W \to X$. (Suppose $i$ is l.c.i. so $i^!$ makes sense, or use obstruction theories. This is automatic if $C, Y$ are smooth). Then Gysin pullback and pushforward commute, so $i^! f_* [D] = f'_* i'^![D] \in A_*(C)$. This is one interpretation of your formula, but beware that $i'^!$ and $i^!$ may differ by the "excess intersection formula" if $f$ isn't flat.

If you're just interested in verifying that the intersection numbers agree, those are given by pushing forward to a point.