Let $A^\ast$ be an algebraic oriented cohomology theory (i.e. it is equipped with certain push-forwards for projective morphisms of smooth varieties over the base field $k$; see section 2 of http://www.math.uiuc.edu/K-theory/0535/orient.pdf for more detail); let $f:Y\to X$ be a finite morphism of smooth varieties whose degree is $d$. I would like to prove the following conjecture: if $f^\ast h=0$ for $h\in A^\ast (X)$, then $d^lh=0$ for some $l>0$ (one cannot take $l=1$ here when $A^*$ is the K-theory). To this end it suffices to verify that $f_\ast 1_Y-d$ is nilpotent in $A^0(X)$ (since $f_\ast f^\ast h=f_\ast 1_Y h$ by the property (v) in the reference cited). It seems sufficient to prove the latter for $A^\ast$ being the algebraic cobordism (as defined by Levine and Morel), since this is the universal oriented cohomology theory.

I would like to say that  $f_\ast 1_Y-d$ is supported in codimension 1. Does $A^0(X)$ possess a multiplicative coniveau filtration? If $f$ is generically etale, then I can use the fact that $f'_*\ast 1_{Y'}=d$ for $f':Y'\to X'$ being the (etale) morphism of generic points; Levine proves this in his cobordism book. 

Is there a better way to prove my conjecture (that would work even if $f$ is not generically etale)?