Let $f:X\to Y$ be a proper morphism of normal varieties (smooth as DM stacks, but I only care about the coarse spaces). The map $f$ is generically finite, but not flat (so no hope of smoothness and Grothendieck-Riemann-Roch applying) and I have a fairly detailed understanding of the fiber over any point, images of $f$ restricted to divisors and so forth.

Now, take a divisor $D$, and identify it with an invertible sheaf in the standard way. I'm looking for a way to compute the first Chern class of $f_*D$ on $Y$.